La 

367 
34 


UC-NRLF 


SEM 


L.   P.    SHIDY 


FORMULAS  AND  TABLES 


ARCHITECTS  AND  ENGINEERS 


CALCULATING  THE  STRAINS  AND  CAPACITY 


OP 


STRUCTUKES   IN   IRON  AND   WOOD, 


BY 


F.   SCHUMANN,  C.  E. 


ILLUSTRATED  WITH   MORE  THAN  THREE   HUNDRED  DIAGRAMS,   DESiGSEL   .uSi> 
ENGRAVED  ESPECIALLY  FOR  THIS  WORK  BY  J.  C.  LYONS. 


WASHINGTON   CITY: 

WARREN   CHOATE   &   CO. 

1873. 


Entered  according  to  Act  of  Congress,  in  the  year  1873,  by 

F.    SCHUMANN, 
In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


IN  ME?/!Of?IAM 

KL 


STEREOTYPED  BY 

tt'OILL  A  WITHEROW, 

WASHINGTON,  D.  C. 


THIS   VOLUME 

is 
RESPECTFULLY  DEDICATED 

TO 

A.     B.     MULLETT, 

gUPERVISING  ARCHITECT  OF  THE  U.  8.  TREASURY  DEPARTMENT, 

BY    THE    AUTHOR. 


(iii) 


CONTENTS. 


EEEATA. 

On  page  4,  10th  line  from  bottom,  read  -— -  instead  of  30. 

On  page  4,  10th  line  from  bottom,  read   10.0036  instead  of 
10.036. 

a 
On  page  4,  14th,  15th,  and  16th  lines  from  bottom,  read  — — - 

instead  of  a. 

On  page  32,  Fig.  70,  insert  I  =  distance  between  supports. 

On  page  34,  Fiq.  72,  insert  I  =  distance  between  supports. 

On  page  34,  Fig.  74,  insert  I  =  length  of  beam. 

On  pages  38  and  39  w  =  total  weight  of  beam  between  supports. 

On  page  39,  5th  line  from  top,  read  1099000  instead  of  1000000. 

On  page  39,  5th  line  from  top,  read  1754  instead  of  1757. 

On  pages  144  and  145,  in  formulas  for  Hn,  change  places  of  last 
minus  sign  with  foregoing  plus  sign.     (See  13th  line  from  top.) 

Page  145,  lines  1  to  7  from  bottom,  ~|  Change  places  of  Cand  T 

Page  146,  lines  1  to  3  from  top,         >     under  strains  in  Figs. 

Page  146,  lines  13  to  22  from  top,    J      225,  226,  227,  and  228. 

On  page  149,  1st  line  from  bottom,  read  — --  — — —  instead 

/„/, 

of- 


N 

On  page  197,  7th  line  from  bottom,  read  3.14159  instead  of 
1.14159. 
On  page  204,  1st  line  from  bottom,  read  A-\-  A,  instead  ofAA,. 


Static  and  moving  loads  on  bridges  of  wrought  iron.,,...  192,  193 
(vii) 


CONTENTS. 


PAGES. 

Summary  of  definitions  and  general  principles 1-5 

^Moments  of  inertia  and  resistance  of  various  sections....        5-25 

Strength  of  materials,  &c 26-29 

Resistance  to  cross-breaking  and  shearing 29 

Capacity  and  strength  of  beams 29-99 

Pressure  on  supports 100-102 

Compressive  strains  and  pressure  on  supports 102 

Sloping  beams,  rafters,  &c 102-103 

Resistance  to  crushing 103 

Strength  of  columns,  pillars,  and  struts 103-110 

Parallelogram  of  forces Ill 

Strains  in  frames 112-114 

Strains  in  boom  derricks 114-115 

Strains  in  trusses.... 115-121 

Strains  in  trussed  beams 122-125 

Strains  in  trusses  with  parallel  booms 126-146 

Strains  in  parabolic  curved  trusses 147 

"Bow-string  girders" 147-153 

Capacity  and  strength  of  parabolic  arched  beams  or  ribs 

originally  curved 153, 154 

Strains  in  a  polygonal  frame 154, 155 

Strains  in  roof  trusses 156-178 

Pressure  of  wind  on  roofs 178, 179 

Pressure  of  snow  on  roofs 180 

Tie  rods  and  bars 181, 183 

Joints  or  connections  in  iron  constructions 184-186 

Dimensions  of  bolts  and  nuts 187, 188 

Compound  strain  in  horizontal  and  sloping  beams 188-190 

Weight  of  moving  loads 191 

Static  and  moving  loads  on  bridges  of  wrought  iron 192, 193 

(vii) 


Vlll  CONTENTS. 

MISCELLANEOUS. 

PAGES. 

Geometry 197-201 

Center  of  gravity  of  planes 202-204 

Trigonometrical  formulas 205 

Trigonometrical  functions 206-217 

Circumference,  area,  and  cubic  contents  of  circles 218-223 

Specific  gravities  of  materials 224-226 

Weight  of  a  superficial  inch  of  wrought  and  cast  iron...  227 

Weight  per  square  foot  of  metals 228 

Weight  of  a  lineal  foot  of  flat  and  square  bar  iron 229-233 

Weight  of  a  lineal  foot  of  rolled  round  iron 234 

Weight  of  bolts,  nuts,  and  heads 235-237 

Weight  of  materials  used  in  building 2381 

Divisions  of  a  foot  expressed  in  equivalent  decimals 239 

Table  for  comparing  measures  and  weights  of  different 

countries 240,  241 

To  cut  the  strongest  and  stiffest  beam  from  a  log 242 


FORMULAS  AND  TABLES 

FOR 

ARCHITECTS  AND  ENGINEERS. 


Summary  of  Definitions  and  General  Principles. 

EXTERNAL  FORCES  are  those  forces  (loads,  &c.)  which  cause  or 
tend  to  cause  the  rupture  of  a  structure. 

INTERNAL  FORCES  are  those  forces  which  resist  the  external 
forces;  when  one  balances  the  other,  the  structure  is  said  to  pos 
sess  Stability 

EXTERNAL  FOKCKS.  INTERNAL  FORCES. 

Compressive  strain.  Resistance  to  Compression. 

Tensional  strain.  Resistance  to  Tension. 

Shearing  strain.  Resistance  to  Shearing. 

Cross-breaking  strain.  Resistance  to  Cross-breaking. 

COMPRESSION  causes  the  material  to  fail  by  crushing  or  buck 
ling,  or  both. 

RESISTANCE  to  direct  Crushing:  In  case  pillars,  blocks,  struts, 
or  rods,  along  which  the  strains  act,  are  not  so  long  in  propor 
tion  to  their  diameter  as  to  have  a  tendency  to  give  way  by 
bending  sideways.  This  includes — 

Stone  and  brick  pillars  and  blocks,  of  ordinary  proportions; 
Pillars,  struts,  and  rods  of  cast  iron,  in  which  the  length  is 
not  more  than  five  times  the  diameter,  approximately; 

Pillars,  struts,  and  rods  of  wrought  iron,  in  which  the  length 
is  not  more  than  ten  times  the  diameter,  approximately ; 

Pillars,  struts,  and  rods  of  dry  timber,  in  which  the  length  is 
not  more  than  twenty  times  the  diameter. 
Let  W '==  Crushing  load  in  Ibs. 

C=  Ultimate  resistance  of  material  to  crushing   in 

Ibs.  per  square  inch. 
A  =  Sectional  area  of  pillar,  &c.,  in  square  inches. 

Then  will  TP  =  A  X  C;  and  A  =  -^- 

C 

TENSION,  causes  the  material  to  be  torn  asunder. 
(1) 


AND   GENERAL   PRINCIPLES. 

Resistance  of  bars,  &<?.,  to  teaiing:  the  ultimate  strength  of  a 
b«ir  (co  :te.fcring)  is :  whssa 

T=  Ultimate  resistance  of  the  material  to  tearing,  in 

Ibs.  per  square  inch. 
W=  Tearing  load  in  Ibs. 
A  =  Sectional  area  of  bar,  in  square  inches. 

W 
Then  will  TF=  A  X  T;  and  A  =  — 

SHEARING  causes  the  fibres  of  the  material  to  be  shorn  by  each 
other ;  when  a  bolt  pulls  out  of  its  nut,  the  threads  of  the  screw 
are  said  to  be  sheared. 

Resistance  of  bars,  bolts,  &c.,  when  sheared  at  one  place,  is: 
when 

S  =  Ultimate  resistance,  of  material  to  shearing,  in 

Ibs.  per  square  inch. 
W=  Shearing  load  in  Ibs. 
A  =  Sectional  area  of  bar,  &c.,  in  square  inches. 

W 

Then  will  W=  A  X  8;  and  A  =  -~ 

o 

CROSS-BREAKING  a  beam,  &c.,  supported  at  one  or  both  ends, 
with  a  force  at  right  angles  to  its  length,  sufficient  to  cause  rup 
ture,  is  said  to  be  broken  across. 

Resistance  to  cross-breaking  is  the  resistance  of  the  material 
to  compression,  tension,  and  shearing  combined ;  •. 

The  flanges  or  booms,  in  beams  or  trusses,  resist  the  bending 
moment,  or  moment  of  rupture. 

The  web  or  braces,  in  beams  or  trusses,  resist  the  shearing 
forces. 

CAPACITY  means  the  load  or  pressure  a  structure  is  capable  of 
sustaining  with  safety. 

DEFLECTION  is  the  displacement  of  a  beam  from  a  horizontal, 
caused  by  its  own  weight  or  the  applied  load,  or  both. 

CAMBER  is  given  a  beam  to  counter  balance  the  deflection,  so 
that  the  beam  may  be  horizontal  when  loaded ;  the  camber  should 
be  equal  to  the  computed  deflection. 

To  find  the  effect  of  combining  several  loads  on  one  beam,  whose 
separate  actions  are  known:  for  each  cross  section,  the  shearing 
force  is  the  sum  of  the  shearing  forces,  and  the  bending  moment 
the  sum  of  the  bending  moments,  which  the  loads  would  produce 
separately. 

When  a  load  on  a  structure  is  partly  static  and  partly  moving, 
multiply  each  part  of  the  load  by  its  proper  factor  of  safety,  and 


DEFINITIONS   AND   GENERAL   PRINCIPLES. 

add  together  the  products  :  the  sum  will  be  the  load  to  which  the 
structure  is  to  be  adapted. 

For  a  Bridge  with  two  platforms,  one  carrying  a  road  and  the 
other  a  railway,  those  two  loads  are  to  be  combined. 

Of  Iron  Ties,  Struts,  and  Beams. 

In  designing  ordinary  structures  of  wrought  iron,  it  saves  time 
and  expense  to  use  iron  bars  of  such  forms  of  cross  section  as  are 
usually  to  be  met  with  in  the  market.  An  engineer  should 
avoid  introducing  new  sections  for  bars  into  his  designs,  except 
when,  by  so  doing,  some  important  purpose  is  to  be  served,  or 
some  decided  advantage  to  be  gained.  The  most  common  forms 
of  rolled  bars  is  shown  by  the  following  enumerated  figures : 


No.  of 
figure. 

Name  of  Form. 

Applicable  for  — 

4 

Square  iron  

Ties. 

13 

Round  iron  

Ties,  bolts,  and  rivets. 

2 

Flat  iron  

Ties. 

29 

I  or  double  T-iron 

Beams  rafter*'  and  struts 

30 

Channel  iron  '.  

Rafters  and  struts. 

37 

T-iron 

Rafters  and  struts 

47 

L  or  angle  iron  

Various  purposes. 

1 

Deck  Beam  

Beams  and  rafters 

Avoid  the  use  of  cast  iron  for  ties,  also  trussed  cast-iron  beams. 

When  a  member  of  a  structure  acts  alternately  as  a  strut  and 
as  a  tie,  it  must  have  sufficient  total  sectional  area,  and  sufficient 
stiffness,  to  resist  the  greatest  compressive  strain  that  can  act,  and 
sufficient  effective  sectional  area  to  resist  the  greatest  tensional 
strain  which  can  act  along  it. 

Let  all  pins,  bolts,  rivets,  &c.,  exposed  to  a  shearing  strain, 
fit  tight  in  its  hole  or  socket. 

Roof  trusses,  the  divisions  of  a  rafter,  and  also  the  struts,  may 
be  considered  as  hinged  at  the  ends. 

In  members  under  a  compound  strain,  as  for  instance  a  trussed 
beam  with  a  distributed  load,  be  careful  to  take  into  account  the 
additional  compression,  caused  by  the  bending  moment. 

The  best  distribution  of  the  material  in  a  section  of  a  cast-iron 

T          0  s  Q 

beam,  for  cross-breaking,  is  that = ;  or  — —  =  — - 

s          s/  '         s  T 

When  T=  Ultimate  resistance  of  the  material  to  tension. 

C=  Ultimate  resistance  of  the  material  to  compression. 

s  —  Distance  from  neutral  axis  to  most  extended  fibres. 

s,  =  Distance  from  neutral  axis  to  most  compressed  fibres. 

That  is,  the  fibres  most  in  tension  should  be  nearest  the  neutral 

axis  of  beam. 


DEFINITIONS   AND   GENERAL   PRINCIPLES. 


In  wrought-iron  beams,  the  section  may  be  made  alike  above 
and  below  the  neutral  axis. 

THE  MODULUS  OF  RUPTURE  should  be  applied  to  beams  with 
full  section,  or  beams  with  continuous  web  only  ;  for  all  open  web 
beams,  or  beams  with  very  thin  web,  the  resistance  of  the  mate 
rial  to  crushing  or  tearing,  respectively,  must  be  used  instead. 

EXPANSION  AND  CONTRACTION  of  long  beams,  which  arise  from 
the  changes  of  atmospheric  temperature,  are  usually  provided  for 
by  supporting  one  end  of  each  beam  on  rollers  of  steel  or  hard 
ened  cast  iron.  The  following  table  shows  the  proportions  in 
which  the  length  of  a  bar  of  certain  materials  is  increased  by  an 
elevation  of  temperature  from  the  melting  point  of  ice  (32°  Fahr., 
or  0°  Centigrade)  to  the  boiling  point  of  water  under  the  mean 
atmospheric  pressure,  (212°  Fahr.,  or  100°  Cent.;)  that  is,  by  an 
elevation  of  180°  Fahr.,  or  100°  Cent,: 


METALS. 

Brass 0.00216 

Bronze 0.00181 

Copper 0.00184 

Cast  iron 0.00111 

Wrought  iron 0.00120 

Tin 0.00225 

Zino 0.00294 

Lead 0.00290 


EARTHY  MATERIALS. 

Brick,  common 0.00355 

Brick,  fire 0.00050 

Cement 0.00140 

Glass,  average 0.00090 

Granite 0.00085 

Marble 0.00087 

Sandstone 0.00105 

Slate 0.00104 


Reference. 
Let  u  —  Value  given  in  above  table,  for  a  certain  material. 

I  —  Length  of  a  bar  at  0°  Centigrade, 

and      Zx—  its  length  at  a  given  number  of  degrees  Centigrade. 
a—  Given  number  of  degrees,  at  which  I,  is  required. 
A  =  Superficial  area  of  a  plate ; 
and    A,—  its  area  at  a  given  number  of  0°  C. 

B  =  Cubic  contents  of  a  body, 
and    .#,=  its  contents  at  a  given  number  of  0°  C. 
Then  will  I,  =   I  (1  +au); 
A,  =  A(l  +  2au)', 
B/  =  B  (1  -j-  3  a  u). 

Example :  A  bar  of  wrought  iron  2  inches  square,  is  10  feet 
long  at  a  temperature  of  0°  Centigrade ;  what  is  its  length  at  an 
increased  temperature  of  30°  ? 

Ans :  I,  =  10  (1  +  30  X  0.00120)  =  10.036  feet. 

THE  NEUTRAL  Axis,  in  a  cross  section  of  a  beam,  is  that  layer  of 
fibres  which  are  neither  in  compression  or  tension,  by  the  action 
of  a  load  on  the  beam  ;  it  always  passes  through  the  centre  of 
gravity  of  the  section :  provided  the  limits  of  elasticity  of  the 
material  is  not  exceeded.  A  beam  supported  at  both  ends,  with 
a  load  acting  perpendicular  between  the  supports,  will  cause  the 
fibres  above  the  neutral  axis  to  be  compressed,  and  those  below, 
extended:  the  farther  from  the  fibres  to  the  neutral  axis,  the 
greater  the  stress. 


MOMENTS   OF   INERTIA  AND   RESISTANCE.  5 

Under  MOMENT  OF  INERTIA  of  a  cross  section,  may  be  under 
stood  :  an  internal  force  at  rest ;  a  static  force  resisting  an  exter 
nal  force;  it  means  the  sum  of  all  the  area  elements,  multiplied 
by  the  square  of  their  perpendicular  heights  from  the  neutral 
axis  of  the  section.  The  moment  of  inertia,  which  we  have 
denoted  with  I,  depends  on  the  form  and  dimensions  of  the  cross 
section,  and  is  expressed  in  square  inches. 

MOMENT  OF  RESISTANCE  of  a  cross  section  is  that  static  force 
resisting  an  external  force  of  compression  or  tension  ;  it  is  equal  to 
the  moment  of  Inertia  divided  by  the  distance  of  the  most  ex 
tended  or  compressed  fibres,  respectively,  from  the  neutral  axis. 


MOMENTS  OF  INERTIA  AND  RESISTANCE  OF  VARIOUS 
SECTIONS. 

To  find  the  moment  of  inertia  of  any  given  cross  section — 
FIRST.  Divide  the  section  into  as  many  simple  figures  as  possi 
ble.     (See  diagram,  fig.  1.) 

SECOND.  Find  the  moment  of  inertia  of  each  of  the  simple  figures 
about  its  own  neutral  axis,  and  insert  the  value  in  the  following 
formula : 

Reference. 

Letters  A,  A/t  A//t  =  area  of  simple  figure,  respectively;  and 
d,  d/t  d//t  =  its  distance  from  its  centre  of  gravity 

to  that  of  the  whole  section. 

i  Vi  V/»  =  moment  of  inertia  of  simple  figures,  re 
spectively. 

For  neutral  axis  see  centre  of  gravity. 
Fig.  1.  B         y 


Formula. 

1=  (i  +  dU)  +  (v  +  d,*A,)  + 
(i//  -{-  djfAji)  +  <fcc.,  =  moment 
of  inertia  of  whole  section. 


MOMENTS  OF  INERTIA  I  AND  MOMENTS  OF  RESISTANCE  — • 

I 

Reference. 

m  —  n  =  neutral  axis  of  section. 
r  =  radius. 
s  =  distance  from  neutral   axis  to  most  compressed  or 

extended  fibres. 
6,  h,  &c.  =  dimensions. 
A  =  area. 


MOMENTS   OF   INERTIA  AND   RESISTANCE. 


No.  of  Figure. 


Form  of  Section. 


2  and  3 


It 

Tfo 


LJL7I' 


JHfr 


m 


n 


771 


MOMENTS   OF   INEETIA  AND    EESISTANCE. 


Moment  of  Inertia  7. 


Moment  of  Resistance- 


=  Tv  Ah* 


bh* 


h* 

6 


12 


h*  - 


Qh 


12 


1/2 


MOMENTS   OF   INERTIA  AND    RESISTANCE. 


No.  of  Section. 


VI. 


No.  of  Figure, 


Form  of  Section. 


VII. 


VIII. 


10 


IX. 


11 


12 


MOMENTS   OF   INERTIA  AND   RESISTANCE. 


Moment  of  Inertia  /. 


Moment  of  Resistance  - 


A 


A 


A1 


A 


jly    6^3    = 


10 


MOMENTS  OF   INERTIA  AND   RESISTANCE. 


No.  of  Section.    No.  of  Figure. 


XI. 


13 


Form  of  Section. 


XII. 


14 


XIII. 


XIV. 


15 


16 


XV. 


17  and  18 


MOMENTS   OF   INERTIA  AND  RESISTANCE. 


11 


Moment  of  Inertia  /. 


Moment  of  Resistance  JL 


}  TT  r»  =  J  Ar 


J 
i* 


s  =  0.576/fc  =  (1 A_  )  h 


12 


MOMENTS   OF   INERTIA  AND  RESISTANCE. 


No.  of  Section.    No.  of  Figure. 


XVI. 


19 


Form  of  Section. 


XVII. 


20 


XVIII. 


21 


XIX. 


XX. 


22 


23 


TTU 


-  - 


MOMENTS   OF   INERTIA  AND    RESISTANCE.  13 


Moment  of  Inertia  /. 


Moment  of  Resistance  — 


-  bh*  =  &  Alt 


15    -  •    10 


MOMENTS   OF   INERTIA  AND   RESISTANCE. 


No.  of  Section. 


No.  of  Figure. 


Form  of  Section. 


XXI. 


24 


XXII. 


XXIII. 


25 


26 


?  /* 

/ 


XXIV. 


27 


XXV. 


28,  29,  and  30 


MOMENTS   OF   INERTIA  AND   RESISTANCE.  15 


Moment  of  Inertia  /. 


Moment  of  Resistance  - 


I A  [i  V  cos*u  +  Jf  A2  sin*v] 


A  [A2  cos2?;  +  V  siri*v] 


12 


I 
*/, 


16 


MOMENTS   OF   INERTIA  AND   RESISTANCE. 


No.  of  Section. 


No.  of  Figure. 


Form  of  Section. 


XXVI. 


31 


XXVII. 


XVIII. 


32 


33 


fel   kM 


XXIX. 


34 


XXX. 


35 


MOMENTS   OF   INERTIA  AND    RESISTANCE.  17 


Moment  of  Inertia  /. 


12 


Moment  of  Resistance  . 


b  (h*  -  A/) 
~~~~ 


1 

~o/r 


18- 


MOMENTS    OF    INERTIA  AND    RESISTANCE. 


No.  of  Section. 


XXXI. 


XXXII. 


No.  of  Figure. 


36  and  37 


38 


Form  of  Section. 


FF^li 

.'  V  \  .'  A    . 


XXXIII. 


XXXIV. 


XXXV. 


39 


41 


-^~"~^j3i 


MOMENTS   OF   INERTIA  AND   RESISTANCE.  19 


Moment  of  Inertia  /. 


Moment  of  Resistance  - 


A  (bh*  +  W) 


6/1 


A  (*6»  +  »„&,*) 


hb*  + 


66 


A  l(**S  —  3625  A 


A  t^/4 


6s—  &/ 


(A—  Z))  4s]  —  0.049  Id* 


IT 


20 


MOMENTS   OF   INERTIA  AND   RESISTANCE. 


No.  of  Section. 


Nj.  of  Figure. 


Form  of  Section. 


XXXVI. 


42 


XXXVII. 


XXXVIII. 


XXXIX. 


XL. 


44 


45 


46, 47,  and  48 


MOMENTS   OF    INERTIA  AND   RESISTANCE. 


21 


Moment  of  Inertia  I. 


Moment  of  Resistance  — - 
s 


-  V) 


A 


__ 

1*7 


12(M"—  6, 


22 


MOMENTS   OF   INERTIA  AND    RESISTANCE. 


No.  of  Section 


No.  of  Figure. 


Form  of  Section. 


XLI. 


49,  50,  and  51 


XL1I. 


XLIII. 


52 


53 


-i 


XLIV. 


XLV. 


55 


MOMENTS   OF   INERTIA  AND   KESISTANCE. 


23 


Moment  of  Inertia  /. 


(bh*  -  6  A2) 2  -  *&/*&/ 


12 


^  r*  \X3~0.5413r4 


Moment  of  Resistance  — 


(bh*-  b^/)'*-  4bhb/h/(h-h/)'> 


=  O.G381 


=  0.5413  (r*— r/) 


=  0  6381  (?•*  —  r/) 


MOMENTS   OF   INERTIA   AND    RESISTANCE. 


No.  of  Section.    No.  of  Figure. 


XLVI. 


XLVII. 


XLVIII. 


56 


57 


58 


Form  of  Section. 


^ 
T  I 


i 

""7 

-'A 


XLIX. 


59 


\        I 


' 


L. 


60 


MOMENTS   OF   INERTIA  AND   RESISTANCE. 


25 


Moment  of  Inertia  J. 

Moment  of  Resistance    I  . 
s 

n/  =  number  of  sides. 
^j  w/r4  sin  .  v  (2  -f-  cos  .  v) 

A  n/  r3  s^?l  •  v  (^  4~  cos  •  v) 

n  /  =  number  of  sides. 
b    =  length  of  a  side. 

TV    ^    (3/&2    -}~     J     O2) 

^1 
i    _^  (3^2  ^_  1  12) 

h 

oJfi  —  b/hf  -j-  b/h/fi 

w_w;w 

12 

6A 

2b^hh//     "  " 

/ 

«/ 

7 

«// 

26 


STRENGTH  OF  MATERIALS. 


STRENGTH  OF  MATERIALS,  &o.f 

In  H>s.,  avoirdupois,  per  square  inch  of  cross-section. 


Materials. 

rt  ^j 
o| 

'£"5 

Ultimate  Resistance  to  — 

Modulus  of 
elasticity. 

Tearing. 

Crushing. 

Shearing. 

Oross-br'k 
Modulus  o 
Rupture. 

METALS. 
Brass,  ca^t,  average  

505.7 
533 
524 

537 
540 

18000 
40000 
30000 

10000 

30000 

3;ooo 

00000 
105(10 
13400 
to 
20000 

10300 

9170000 
14230000 
9900000 

17000000 
17000000 
14000000 
to 
22900000 

29000000 

25300000 

15000000 

29000000 
to 
42000000 

720000 
4000000 
13000000 

1000000 
1350000 

"       wire.     

Bronze  or  gun  metal,  (cop 
per  8,  tin  1) 
Copper   cast  

117000 

"         she^t    

"         bolt** 

"         \viro        

Iron  ca^t  avcnvo 

445 
4)54 
to 

450 

112000 
80000 
to 
115000 

27700 

"          various  

"           beams,  average.. 

28800 
17000 
33000 
to 
43500 

38000 

"           solid  rect.  bars, 
various  quailities. 

Iron,  wrought,  average  

481 

05000 

30000 
to 
40000 

50000 

plates  
joints,  d'ble 
riveted. 
Iron,  wrought,  joints,  single 
riveted. 
Iron,    wrought,    bars    and 
bolts. 

hoop,  best  best 
wire  

51000 
35700 

28000 
GdOOO 

to 
70000 
04000 
70000 
to 
100000 
00000 

wire  ropes.... 

Steel,  average  

490 

80000 

"      bars  

100000 
to 
130000 
80000 
3:500 

4(300 
7000 
to 
8oOO 

17000 

0300 
11500 

12000 

7730 
15500 

15000 

"      plates  

Lead,  sheet  

712 

402 
430 

47 
43 

Tin,  cast  

Zinc  

TIMBER,  (well  seasoned  and 
dry.) 
Ash  

9000 
9300 

1400 

12000 
to 
14000 

9000 
to 
20000 

Bamboo  

Beech  

STRENGTH  OF  MATERIALS. 


27 


Materials. 

Weight  of  a 
cubic  foot. 

Ultimate  resistance  to  — 

If 

Tearing. 

Crushing. 

Shearing 

Cross-br'k 
Modulus  o 
Rupture 

|| 

TIMBER  —  Continued. 
Birch         

44 
80 

33.4 

34 

74.5 
37 

37 
33 

52 
47 
52.5 
44 
62 
35 

49 
52.5 

47.4 

15000 

200O( 
100O( 

to 
1300< 
140iK 

120CX 

to 
14000 
12401 

9QOC 

to 
10*  -00 
25000 
20000 
23400 
1COOO 
ll.XOO 
8000 
to 
21800 
10600 
10000 
o 
19800 

6400 

10300 

5300 

10300 

19-)00 

5375 
to 
6200 

5900 

5570 

11000 
7300 

9000 
9900 
8200 

6500 
10000 

7700 
6100 
6000 
5300 
5400 
12000 
12000 

11000 
6500 
4000 

550 
to 
800 
1100 
1700 
417 
to 
612 

11701 
1086 

6004 

to 

2700 
7  UK 
to 
9541 
990, 
to 
1  'JjOt 
50.  < 
to 
10001 

1735! 
1  1  ..00 
1200 
10000 

10000 
to 
13(00 
87  0< 

10601) 

9600 
12000 
to 

17460 
6600 

1645000 
1140000 

700000 
to 
1340000 

146  000 
to 
1900000 
UOO  00 
to 
1800000 
900000 
to 
1  .".60000 
10-tUOOO 

1255000 

1200000 
to 
1750000 

21  50000 
2400000 

Box....        .... 

Chestnut  

Elm 

1400 

Ebony,  West  Indian.... 
Fir  Red  Pine  .. 

500 
to 
8(iO 
600 

970 
to 
1700 

"    Spruce  

"    Larch 

Hickory  

Hornbeam   

Lance  wood  
Locust  

Lignum  vitse  

Mahogany  

Maple...... 

2300 

Oak,  British  

"      Dantzic  

"     American  white.... 

42 
54 
346 

29 
37 

48 

62.5 

18000 
10250 
11500 
15000 
13000 
15000 

red  

Pine,  American,  white  

"               yellow  
Sycamore  
Teak,  Indian  

Water  gum  

Walnut  

40 
25 
50 

125 

135 
37.5 

loo 

8000 
14000 
8000 

280 
to 
300 

Willow,  various  

Yew  

STOXES,  (natural   and  arti 
ficial.) 
Brick,  weak  red  

"      strong  red  

"      fire  

"      work  

Cement  

89 

280 
to 
300 

STRENGTH  OF  MATERIALS. 


Materials. 

Weight  of  a 
cubic  foot. 

Ultimate  resistance  to  — 

Modulus  of 
elasticity. 

Tearing 

Crushing. 

Shearing. 

>oss-br'k. 
•Jodukis  ot 
Rupture. 

STONES—  Continued. 
Chalk    

145.5 
173 

168 

118 
9400 

330 

8000000 

13000000 
to 
16000000 

Glas=! 

Granite  

5500 

2360 

1100 
5000 

Limestone  marble  

172 

to 
11000 
5500 
4000 
to 
4500 

About 
4-10  cut 
stone. 
5500 
3300 
to  4400 

"           granular  

197 

100 
to 
170 
50 

109 
116 

Rubble  masonry  
Sandstone,  strong  ") 

"          ordinary             ( 

144 

"          weak  ) 

Slate  

178 

9600 
to 
12800 

25000 
14000 
6300 
4200 
5200 
7700 

MISCELLANEOUS. 
Flaxen  yarn        •  

Hempen  ropes... 

Hide,  ox  undressed    ... 

Leather  ox 

Silk  fibre  

Whalebone  

MODULUS  OF  RUPTURE  R. 

According  to  Professor  Rankine,  the  modulus  of  rupture  is 
eighteen  times  the  weight  that  is  required  to  break  a  bar  of  a 
given  material  one  inch  square  (section)  and  one  foot  between 
supports,  the  weight  concentrated  at  the  middle. 

MODULUS  OF  ELASTICITY  E 

Is  that  power  (in  Ibs.  generally)  through  which  a  prismatic  body 
of  a  given  material,  of  section  =  1,  is  assumed  to  be  extended 
double  its  length,  or  compressed  to  0. 

Let  A  =  Sectional  area  of  a  rod  of  the  material. 

W=  Weight  or  power  in  Ibs.,  which  causes  the  extension 

or  compression  of  the  rod. 

I  =  Length  in  inches  of  rod  before  W  is  applied. 
Y  =  The  extension  or  compression  of  the  rod  in  inches, 
caused  by  W. 

Wl  W 


RESISTANCE  TO  CROSS  -BREAKING  AND  SHEARING.  29 

FACTORS  OF  SAFETY  k. 
The  ultimate  resistance  of  material  should  be  divided  by — 

A  WrolgifttI1eonnd         For  Proof  strenSth-     Foi>  Working  Stress. 

Steady  load 2     

Moving  load 4  to    6 

Cast  Iron. 

Steady  load 2  to  3     3  to    4 

Moving  load 6  to    8 

Timber. 
Average 3     8  to  10 


RESISTANCE  TO  CROSS-BREAKING  AND  SHEARING. 
CAPACITY  AND  STRENGTH  OF  BEAMS. 

Reference. 

A  =  Area  of  cross-section  of  beam. 
D  —  Deflection  of  beam  from  a  horizontal. 
E  =  Modulus  of  elasticity. 

J=  Moment  of  inertia  of  cross  section. 
M—  Maximum  moment  of  rupture,  or  bending  moment. 
R  =  Modulus  of  rupture. 

S  =  Vertical  shearing  force. 

V  =•  Pressure  on  supports. 
W=  Capacity  or  weight  of  load, 
c,  d,  I  ==  Dimensions  in  units  of  length. 

k  =  Factor  of  safety. 

w  =  Weight  of  load  per  unit  of  length. 

=  Moment  of  resistance  of  cross-section. 

I 

R       I 

For  the  stability  of  a  beam :    M=.  K  = . . 

k        s 

The  web  of  a  metal  beam  must  have  sufficient  area  to  resist  the 

shearing  force  8;   that  is,  A  =  -rrr-: : : 

Ultimate  resistance  to  shearing. 

The  weight  of  the  beam  must  be  added  to  W,  except  in  small 
beams,  under  60    Ibs.  per  lineal  foot,  when  it  may  be  disregarded. 

[NOTE. — Always  use  the  same  units  of  dimensions  or  weight.] 


30  RESISTANCE   TO   CROSS  BRKAKING  AND   SHEARING. 


No.  of  Figure. 


Manner  of  loading  and  fastening  beams. 


61 


.    -P  "   " "    V  x 


62 


63 


64 


. 


If 


' 


o 
p 


xim 
ent 
ure 


W.I 


- 


W. 


W. 


5.333 


ity 
sec 


K 

I 


4     K 

•~ 


5.333—- 


K-  ------    -  -------  - 


.65 


r"V 


RESISTANCE   TO   CROSS  BREAKING   AND   SHEARING. 


31 


Maximum  deflec 
tion  D. 

Distance  from 
A  to  point  of 
maximum/). 

Shearing  force  S. 

Pressure  on  sup 
ports  V. 

W       I3 

.EM'  3 

J 

At  any  point. 
W 

W 

W     Is 
E  I'  8 

I 

At  any  point. 
w  .d 

'  W 

W     p 

-£    lO'ls" 

z 

2 

At  any  point. 
IF 

~Y 

V  -  V  -  W 

1/1  -     ^2--^ 

IF  Z3 
"J       0.00931 
J-j.  J. 

0.553J 

1  TT.-L 

".-".-?- 

&     W     P 

2 

"T 

At  any  point, 
d<d/; 

'  (i-0 

fi-F._i 

8  E.I'  48 

32  RESISTANCE   TO   CROSS-BREAKING   AND   SHEARING: 


66 


68 


69 


Manner  of  loading  and  fastening  beams. 


Maximum  mo 
ment  of  rup 
ture  M. 


'\ 


•/MM 


A 

^  <—rl- 


A 


w|< \ \ — 2- — - 

^ 


\-L 

r*-f — •*• — i 


± 


^        --------  £_  ------  ^ 


- 


IF.— 
12 


W  .1  + 


W 
tion. 


Capaci 
any 


8.— 


12"T- 


-.K 


RESISTANCE    TO    CROSS  BREAKING    AND    SHEARING. 


|CQ 

^"S§ 

Maximum  deflec 
tion  D. 

§11 
pr§ 

Shearing  force  5. 

Pressure  on  sup 
ports  F. 

"DQTJH  C 

P 

W       Z3 

j 

ir 

W 

E.I  4.48 

O 

^ 

W  Z3 
0  00r)4 

Q.  572.1 

n  tt      1       '     "                    ,T\ 

v      v      w 

E.I    ' 

(  8    '       1 

W        P 

l 

'*<** 

E.I'   8.48 

o 

-(!-) 

^ 

(Irr¥)  + 

At  any  point  be 

/  Wl    ^3> 

i 

tween  loads. 

(  —  —  y  -0~  •  )  ~t~ 

^ 

S=  W  .Sl= 

II    1    \\  l  \    n  2 

£/.J.       •) 

~m~  _i_  ~nr      ^  

(\ 

W  4-  wl  +  Wz 

E.I'~3") 

At  any  point  and 

under  any  load. 

£_  }y    12 

V   —  -?-   W 

W         /3         72/2 
1          fj         6j 

I 

1 

E.I     3      I2  '  I2 

Constant  bet,  A  &  IF 

J 

s  —  w  ll 

V2  =  1±-  W 

Constantbet.  Z?&  IF 

3-1  RESISTANCE    TO   CROSS-BREAKING  AND   SHEARING. 


i  Maximum  moment  of 
Manner  of  loading  and  fastening  beam?,   j         rupture  M. 


A 


x:.:v          ^       /C>v         rrK  I 

§1        m    ^l^ 


T^! 


IF./, 


W.I, 


When  ^>^ 


[(4-)'-'.'] 


When  Z  <  Zi  \/8  ; 


RESISTANCE   TO   CROSS-BREAKING  AND   SHEARING. 


35 


•^"c  £ 

Capacity  W  of  any 
section. 

Maximum 
deflection  D. 

sll 
Is! 

Shearing 
force  8. 

Pressure 
on  sup 
ports  V. 

5 

W 

E.I 

K 
h 

a 

2 

W 

w2 

h 

i2 

w  l* 

F1== 

Kl 

~~i~w 

^(1-^) 

11 

2Ti 

1 

i  w 

W  12    tl 

-   8  E.I 

K 

Wl^ 

w 

F!  =  F2  » 

h 

D,-     l 

W 

\  2          3  ' 

2  (1  +  ^i)    g 

(4)'-'.- 

w  .l±  or 

w.-JL- 

W 

The  greater 
value  to  be 

2 

taken. 

2(1+21,)   K 

'"' 

36  RESISTANCE   TO   CROSS- BREAKING  AND   SHEARING. 


Manner  of  loading  and  fastening  beams. 


Maximum  moment  of 
rupture  M. 


When  c?  >(Z  —  c)  ; 


w- 


W- 


78 


1^  («!+«,)] 


RESISTANCE  TO   CROSS-BREAKING  AND   SHEARING. 


37 


Capacity  W  of  any 
section. 

MAximum 
deflection  D. 

S'oq 
lag 

§a| 

1-1 
|^l 

Shearing 
force  8. 

Pressure 
on  sup 
ports  F". 

1           K 

e(l_«_d)A 

2K 

(l-r-d)* 

213            E 

W 

—  -^ 

3  E.I  ' 
IS(1—IJ 

1^(31-1^(1-1^) 

I'2 

P         K 

Vff-y" 

38  EESISTANCE  TO  CROSS-BREAKING  AND  SHEARING. 

EXAMPLE.  —  Capacity  of  wrought-iron  l-shaped  beams;  top  and 
bottom  flange  alike  ;  load  equally  distributed  ;  ends  not  fixed. 

Dimensions  of  Cross-section. 

h  =  Height  =  10  inches. 
b  =  Width  of  flange  —  4  inches. 
t  =  Thickness  of  flange  =  0.8  inches. 
t/  =  Thickness  of  web  =  0.5  inches. 
7t/==  h  —  2t;  b,  =  b  —  t,. 

Distance  between  supports  •=  20  feet  =  240  inches.     Factor  of 
safety  =  3. 

MOMENT  OF  RESISTANCE. 

_Z_         ft/*3—  M/3          4  x  1Q3  —  3.5  X  8.4*  _  0 
V  =  6*  ~  6  X  10 

Capacity  W. 
w  =  (4  X  0.8  X  2  +  8.4  x  0.5)  x  240  X  0.28  =  712.32  Ibs. 

K=*.  ^J*™.  32.09  =  406473.33. 
.     k        s  3 

W=8^--w^8.—~^  --  712.32  =  12836.72  Ibs. 

li  ^41) 

EXAMPLE.  —  Capacity  of  cast-iron  i-shaped  beams;  load  equally 
distributed;  ends  not  fixed;  flange  down. 

Dimensions  of  Cross-  section. 

Let    h  =  Height  =  18  inches. 

b  =  Width  of  flange  =  9  inches. 
t  =  Thickness  of  flange  =  1.25  inches. 
t/  =  Thickness  of  web  =  1  inch. 
7i/  =  h  —  t;  by  =  b  —  t/. 

Area  =  28  square  inches.     Distance  between  supports  =  20 
feet  =  240  inches.     Factor  of  safety  k  =  4. 

MOMENT  OF  RESISTANCE. 

sh,  (h  —  h,Y 


*  L    bh2  —  2b/hh/  +  b,h/ 

-*[- 


2b/hh/  + 
(9  x  182  —  8  x  16.752)2 


9  X  182—  2  X  8  X  18  X  16.75  +  8  X  16.752 
4X9X18X8X  16.75  (18  —  16.75)2 


9  X  182  —  2  X  8  X  18  X  16.75+8  X  16. 


5)2    n 
6.752J 


RESISTANCE  TO  CROSS-BREAKING  AND  SHEARING.  39 

_     rj452256.25_       _135675.00_-]  =  15- 
*  °  L      336.5  336  5      J  . 

• 

Capacity  W. 
w  =  28  x  240  X  0.261  =  1754.  lbs. 

JT^;*      J-  =  -^-.  157  =  1099000. 
k        s  4 

JF=  8  A  -  „  =  8  .  --  ~S"  -  1757  =  34879  lbs- 

For  light  beams  no  attention  need  be  paid  to  weight  of  beam  w. 

CAPACITY  If  OF  ROLLED  I  -SHAPED  BEAMS. 
Load  equally  distributed. 

The  calculations  are  based  upon  (lie  patterns  or  section*  used 
by  the  Phcenixville  Iron  Company.  Practically  this  applies  to 
all  similar  beams  rolled  in  the  United  States,  the  difference  in  the 
profile  of  section  being  slight. 

In  the  following  table  the  factor  of  safety  k  =  2.53: 

Reference. 

W=  Load  in  tons  of  2,000  lbs.,  equally  distributed. 
w  =  Weight  of  beam  in  tons  of  2,000  lbs. 
L  =  Distance  between  supports  in  feet. 
I  =  Distance  between  supports  in  inches. 
iu/=  Weight  per  square  foot  of  floor. 

W,=  Capacity  of  coupled  or  trebled  beams  in  tons  of  2,000  lbs. 
D  =  Deflection  in  inches  at  centre,  between  supports. 
d  =  Distance   between   centres  of  beams,  when  spacing  for 
floors,  in  feet. 


W  W,  r>    W+w       J? 

7.0  tons,     d—  —  --  ,  or  d=  -         -,  D  —  f  —  .  —  . 

L.to,  L.w/  &.L         4b 

K1  =  Constant,  computed  by  formulas.    (See  under  examples.) 


40 


RESISTANCE  TO  CROSS -BREAKING  AND  SHEARING. 


The  rivets  for  coupled  or  trebled  beams  should  be  about  £  inch 

in  diameter,  and  8  inches  apart. 


Trebled  Beams. 


Coupled  Beams. 


W,=  WX  5.33. 


— :JQ — ;  =  17.2  tons.     This  is  also  found  at  the  intersection  of 


Fig.  79. 

Examples  explanatory  of  the  following  Table. 

EXAMPLE.-— What  is  the  capacity  of  a  15-inch  light  beam,  load 
equally  distributed,  distance  between  supports  =  20  feet? 

7-1  /f'8  1     TT7  Kl  K1 

A    =  — r^ — ,  and   W—  — —  ;   for  15-inch  light  beam  -----  = 
-Li  J_j 

345. 19 
20  ~ 
20  feet  and  column  under  capacity  W. 

EXAMPLE. — What  distance  apart  should  9-inch  medium  beams 
be  placed,  the  distance  between  supports  being  20  feet,  and  to 
carry  a  total  load  of  140  Ibs.  per  square  foot  of  floor  surface? 

Ans.  4.4  feet;  being  found  at  the  intersection  of  the  horizontal 
line  from  20  feet  and  the  vertical  column  under  140  Ibs. 

EXAMPLE.-— What  is  the  capacity  of  12- inch  light  beams  trebled . 
load  equally  distributed,  distance  between  supports  =  25  feet? 

Ans.  W  for  12-inch  light  beam  ==  9.19  and  W,  =  W  X  5.33  = 
9.19  X  5.33  =  48.98  tons. 


RESISTANCE   TO   CROSS-BREAKING  AND   SHEARING.  41 


CAPACITY  OF  ROLLED  BEAMS. 

Explanation  of  Tables  for  I  Beams. 

The  first  column  gives  the  distance  between  supports  in  feet. 

The  second  column  gives  the  capacity  in  tons  of  2,000  Ibs., 
equally  distributed. 

The  third  column  gives  the  deflection  in  inches  at  centre  of 
beam. 

The  fourth  column  gives  the  weight  of  beam  in  Ibs.  for  length 
between  supports. 

The  fifth  to  fifteenth  column  (inclusive)  gives  the  distance  in 
feet  that  the  beams  should  be  spaced  from  centre  to  centre,  for 
weight  in  Ibs.,  per  sq.  ft.  of  surface  for  floors. 

Pounds  in  decimals  of  a  ton. 

Ibs.       tons. 

GO  ==  0.03 

70  =  0.035 

80  =  0.0-1 

90  =  0.045 
100  =  0.05 
140  =  0.07 
160  =  0.03 
180  =  0.085 
200  =  0  1 
250  =  0.125 
300  ==  0.15 

In  using  these  beams  for  floors,  with  brick  arching,  the  ends 
resting  on  supports  should  have  a  bearing  of  about  8  inches, 
resting  on  a  cast-iron  plate,  8  X  12  in.  sq,,  by  1  in.  thick. 

Tie  rods  should  be  used  where  floors  are  subject  to  heavy  con 
centrated  moving  loads,  (as  trucks  with  merchandise,  &c.;)  these 
rods  should  be  about  8  times  the  depth  of  beam  apart,  fastened 
about  -J  from  the  bottom  of  beam. 

When  beams  are  used  to  support  walls,  or  as  girders  to  carry 
floor  beams,  and  put  side  by  side  (II,)  they  should  be  fastened  to 
gether  with  cast-iron  blocks,  fitting  between  the  flanges,  so  as  to 
securely  combine  the  two  beams.  The  blocks  may  be  put  about 
the  same  distance  apart  as  the  tie-rods. 


42 


RESISTANCE  TO  CROSS  BREAKING  AND  SHEARING. 


Fig.  81. 


15"  "Heavy  "  Beam.     Weight  per  If.  =  66.66  Ibs. 


Sectional  area =  20.0" 

Moment  of  inertia  /  =  652.42 
Constant  #' =434.95 

K' 
W—  —  . 

L 


s  "S 
5 

a 

fl 
05 

Deflec.  in  inches. 

Weight  in  Ibs. 
1 

Distance  d  bet.  centres  of  beams  in  feet,  for 
weight  in  Ibs.  per  sq.  foot  of  — 

1 
8 

i 

.0 

i 

£ 
o 

| 

GO 
.D 

g 

<n 
.0 

j 

B 

1 

8 

i 

8 

i 

i 

1 

20.1 
17.6 
14.7 
12.8 

io!2 

8.9 
8.0 
7.2 
6.7 
5.9 
5.5 
50 
4.7 
4.2 
3.9 
3.6 
3.4 
3.2 
3.0 
2.8 
2.6 
2.5 
2.3 
2.2 
2.1 
2.0 
1.9 
1.8 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

72.49 
62.13 
54.35 
4832 
43.48 
39.54 
36.24 
33.45 
31.05 
28.99 
27.18 
2558 
24.16 
22.89 
21.73 
20.71 
19.58 
18.91 
18.12 
17.39 
16.72 
16.10 
15.53 
14.99 
14.49 
14.03 
13.59 
13.17 
12.79 
12.42 
12.08 
11.75 
11.43 
11.15 
10.87 

0.037 
0.050 
0.065 
0.084 
0.104 
0.126 
0.150 
0.177 
0.205 
0.236 
0.270 
0305 
0.342 
0.383 
0.426 
0471 
0.515 
0.569 
0.623 
0.677 
0.735 
0.795 
0860 
0.925 
0.994 
1.067 
1.141 
1.219 
1.304 
1.384 
1.473 
1.564 
1.656 
1.754 
1.854 

400.0 
466.6 
5333 
600.0 
666.6 
733.3 
800.0 
866.6 
933.3 
1000.0 
1066.6 
1133.3 
1200.0 
1266.6 
1333.3 
1400.0 
1466.6 
1533.3 
1600.0 
1666.6 
1733.3 
1800.0 
1866.6 
1933.3 
2000.0 
2066.6 
2133.3 
2200.0 
2266.6 
2333.3 
2400.0 
2466.6 
2533.3 
2600.0 
2666.6 

20.9 
17.7 
15.5 
13.5 
12,0 
10.7 
9.6 
8.6 
7.9 
7.1 
6.7 
6.0 
5.6 
5.2 
4.7 
4.4 
4.1 
3.8 
3.6 
3.3 
3.1 
3.0 
2.8 
2.6 
2.5 
2.4 
2.2 
2.1 

22.1 
19  '> 

•79  3 

21.2 
19.6 
16.7 
15.2 
13.5 
12.5 
ll.l 
10.5 
9.4 
8.6 
8.0 
7.4 
6.9 
6.8 
6.0 
5.6 
5.3 
4.9 
4.7 
4.4 
4.1 
39 
3.7 
3.5 
3.3 

18.8 
17.0 
14.9 
13.4 
12.0 
11.5 
9.8 
9.4 
8.3 
7.7 
7.2 
6.7 
6.2 
5.7 
5.3 
5.0 
4.7 
4.4 
4.1 
3.9 
37 
3.5 
3.3 
3.1 
2.9 

LG.9 
15.0 
13.4 
12.0 
10.8 
9.8 
8.9 
8.2 
7.5 
6.9 
6.4 
5.9 
5.5 
5.1 
4.8 
4.5 
4.2 
3.9 
3.7 
3.5 
3.3 
3.1 
3.0 
2.8 
2.7 

21.4 
19.1 
17.6 
15.5 
14.7 
127 
11.8 
10.7 
10.2 
9.2 
8.5 
7.9 
7.4 
6.9 
6.4 
6.0 
5.7 
53 
5.0 
4.7 
4.4 
4.2 
4.0 
3.8 

2-i'o 

19  7 

21.7 
19.7 
17.8 
17.1 
15.1 
14.4 
12.8 
11.9 
11.0 
10.7 
9.8 
9.0 
8.4 
7.9 
7.5 
7.1 
6.6 

e.3 

6.0 
5.7 
5.4 

21.0 
18.8 
17.3 
16.7 
14.9 
13.8 
12.9 
12.0 
11.3 
10.0 
9.9 
9.4 
8.8 
8.4 
7.9 
7.5 
7.1 
6.7 

18.9 
16.7 
15.5 
15.2 
13.4 
12.3 
11.5 
10.7 
10.0 
9.4 
8.8 
8.2 
7.9 
7.4 
7.0 
6.6 
6.3 
6.0 

21.4 
19.8 
18.2 
17.2 
16.1 
15.0 
14.0 
13.3 
12.5 
11.8 
11.1 
10.8 
10.0 
9.5 
9.0 

21.5 
19.9 
18.3 
17.1 
15.8 
14.8 
13.8 
12.9 
12.0 
11.4 
10.7 
10.1 
9.5 
9.1 
8.5 
8.1 
7.7 

RESISTANCE   TO   CROSS  BREAKING  AND   SHEARING. 


43 


Fig.  82. 


15"  "Light"  Beam.     Weight  per  If.  =  51.66  Ibs. 


Sectional  area =    15.5" 

Moment  of  inertia  /  —  517.78 
Constant  K' =345.19 

K' 
W  =•-. 

L 


3 

00 

<D 

Distance  d  bet.  centres  of  beams  in  feet,  for 

o 

a 

c 

o 

00 

weight  in  Ibs.  per  sq.  foot  of  — 

d 

0 

£ 

CO   CJ 

^ 

a 

£3 

|.s 

| 

i 

lip 

| 

.0 

Ul 

.C 

Vl 

.0 

?] 

| 

& 

JS 

j5 

09 

X3 

72 

& 

ft 

1 

^c 

<S 

9 

8 

o 

£ 

§ 

Srj 

o 
•& 

1 

>g 

1 

CM 

1 

6 

57.52 

0037 

310.0 

7 

49.31 

0.050 

361.6 

8 

43.13 

0.065 

413.3 

9 

38.35 

0.084 

4650 

10 

34.50 

0.103 

51  6.6 

23.0 

11 

31  .38 

0.124 

567.3 

22.9 

19.0 

12 

28.76 

0.150 

620.0 

19  T15.9 

13 

26.55 

0.176 

671.  6 

22.7 

^0  4 

164  13.6 

14 

24.  G5 

0.205 

793  ^ 

22.0 

19.5 

17.6  14.0  11.7 

15 

23.01 

0.236 

775.0 

21.9 

19.1 

17.0 

15.3  12.3  10.2 

16 

21.57 

0.269 

806.6 

19.2 

16.8 

14.9 

13.4  10.71    8.9 

17 

20.30 

0  304 

858.3 

17  9 

14.9 

13.2 

11.9     95!    7.Q 

18 

19.16 

OJ341 

830.0 

21.3 

15^2 

13.3 

11.8 

10.6 

8.5  j    7.1 

19 

18.15 

0.381 

981.6 

21.3 

19.1 

13.6 

11.9 

10.6 

9.5 

7.6 

6.3 

20 

17.24 

0.424 

1033.3 

21.5 

19.1 

17.2 

12.3 

10.7 

9.5 

8.6 

6.9 

5-7 

21 

16.43 

0.469 

1085.0 

19.5 

17.4 

15.6 

11.1 

9.8 

8.7 

7.8 

6.2 

5.2 

22 

15.68 

0.515 

1136.6 

20.3 

17.8 

15.8 

14.2 

10.1 

8.9 

7.9 

7.1 

5.7 

4.7 

23 

15.00 

0.565 

1187.3 

21.7 

18.7 

16.3 

14.5 

13.0 

9.3 

8.1 

7.2 

6.5 

5.2 

4.3 

24 

14.38 

0.620 

1240.0 

19.9 

17.1 

14.9 

13.3 

11.9 

8.5 

7.4 

6.6 

59 

4.8 

3.9 

25 

13.80 

0.674 

1291.6 

18.4 

15.8 

13.8 

12.3 

11.0 

7.8 

6.9 

6.1 

5.5 

4.4 

3.6 

2(5 

13.27 

0.732 

1343.3 

17.0 

14.5 

12.8 

11.3 

10.2 

7.2 

6.3    5.6 

5.1 

4.0 

3.4 

27 

12.78 

0.791 

1395.0 

15.7 

13.6 

11.8 

10.5 

9.4 

6.7 

5.9    5.2 

4.7 

3.7 

3.1 

28 

12.32 

0.855 

1446.6 

14.6 

12.5 

11.0 

9.7 

8.8 

6.2 

5.5 

4.8 

4.4 

3.5 

2.9 

29 

11.93 

0.921 

1498.3 

13.7 

11.8 

10.2 

9.1 

8.2 

5.8 

5.1 

4.5 

4.1 

3.2 

2.7 

30 

11.50 

0.989 

1550.0 

12.7 

10.9 

9.5 

8.5 

7.6 

5.4 

4.7 

4.2 

3.8    3.0 

2.5 

31 

11.13 

1.060 

1601.6 

11.9 

10.3 

8.9 

8.0 

7.1 

5.1 

4.4 

3.9 

3.5    2.8 

2.3 

32 

10.78 

1.133 

1653.3 

11.2 

9.( 

8.4 

7.4 

6.7 

4.8 

4.2 

3.7 

3.3    26 

2.2 

33 

10.46 

1.211 

1705.0 

10.5 

9.0 

7.9 

7.0 

6.3 

4.5 

3.9 

3.5 

3.1 

2.5 

2.1 

34 

10.14 

1.292 

1750.6 

9.9 

8.5 

7.4 

6.6 

5.9 

4.2 

3.7 

3.3 

2.9 

2.3 

1.9 

35 

9.86 

1.375 

1808.3 

9.3 

8.( 

7.0 

6.2 

5.6 

4.0 

3.5 

3.1 

2.8 

2.2 

1.8 

36 

9.58 

1.463 

1860.0 

8.8 

7.6 

6.6 

5.9 

5.3 

3.8 

3.3 

2.9 

2.6 

2.1 

1.7 

37 

9.32 

1.553 

1911.6 

8.3 

7.2 

6.2 

5.6 

5.0 

3.5 

3.1 

2.7 

2.5 

2.0 

1.6 

38 

9.08 

1.645 

1963.3 

7.£ 

6.8 

5.9 

5.3 

4.7 

3.4 

2.9 

2.6 

2.3 

1.9 

1.5 

39 

8.85 

1.742 

2015.0 

7!5 

6.5 

5.6 

5.0 

4.5 

3.2 

2.8 

2.5 

2.2 

1.8 

1.4 

40 

8.62 

1.841 

2066.6 

7.1 

6.1 

5.3 

4.7 

4.3 

3.0 

2.6 

2.3 

2.1 

1.7 

1.4 

RESISTANCE   TO   CROSS- BREAKING  AND   SHEARING. 


Fig.  83. 


12"  "Heavy"  Beam.     Weight  per  If.  =  56.66  Ibs. 


Sectional  area =    17.0" 

Moment  o'l  inertia  /  =  373.53 

Constant  K' =  311.28 

K' 
W  =  — . 


•g 

d 

00 

Distance  d  bet.  centres  of  beams  in  feet,  for 

o 

ft  . 

G 

x> 

weight  in  Ibs.  per  sq.  foot  of  — 

ft-g 

d 

d 

"-I 

»«2 

u 

d 

.2 

I'S 

o 

Gj 

6 

o 

I 

§ 

00 

32 

_Q 

OQ 

& 

i 

£ 

A 

J§ 

» 

| 

n 
ft 

C3 
0 

o 

ft  • 

'o 

o 

o 

0 

o 

o 

o 

o 

g 

§ 

§ 

1 

6 

51.88 

0.046 

340.0 

7 

44.54 

0.063 

396.6 

8 

38.70 

0.082 

453.3 

<) 

34.58 

0.105 

510.0 

10 

31.12 

0.131 

566.6 

20.7 

28.29 

0.158 

623  ? 

•'0  •" 

17.1 

12 

25.9^ 

0.188 

680.0 

21.6 

17.2 

14.4 

-iq 

23.94 

0.222 

736.6 

23.0 

20.4 

18.4 

14.7 

12.2 

lo 
14 

22/22 

0  258 

793.3 

22.( 

19.8 

17.6 

15.8 

12.6 

10*.5 

15 

20.75 

0.297 

850.0 

10  7 

17.2 

15.3 

138 

11.0 

9.2 

16 

19*.5( 

0.339 

906.6 

17.4 

15.2 

13.5  12.1 

9.7 

8.1 

17 

0.383 

963*3 

21.5 

15.3 

13.4 

11.9  10.7 

8.6 

7.1 

18 

17/29 

0.431 

1020.0 

21.  P 

19.2 

13.7 

12.0 

10.6    9.6 

7.6 

6.4 

19 

16.38 

0.481 

1076.6 

21.5 

19.1 

17.2 

12.3 

10.7 

9.5     8.6 

6.8 

5.7 

20 

15.61 

0.538 

1133.3 

19.5 

17.:J 

15.  ( 

11.1 

9.7 

8.6 

7.8 

6.2 

5.2 

21 

14.82 

0.592 

1190.0 

20.1 

17.  ( 

15.( 

14.1 

10.0 

8.8 

7.8 

7.0 

5.6 

4.7 

22 

14.1-1 

0.652 

1246.6 

21.4 

18.3 

16.( 

142 

12.8 

9.1 

8.0 

7.1 

6.4 

5.1 

4.2 

23 

13.53 

0.717 

1303.3 

19.0 

16.8 

14."~ 

130 

11.7 

8.4 

7.3 

6.5 

5.8 

4.7 

3.9 

24 

12.9" 

0.786 

1360.0 

18.0 

15.4 

13.5 

12.( 

10-8 

7.7 

6.7 

6.0 

5.4 

4.3 

3.6 

25 

12.4 

0.855 

1416.6 

16.6 

14.2 

12.4 

11.0 

9.r 

7.1 

6.2 

5.5 

4.9 

3.9 

3.3 

26 

11.9 

0.927 

1473.3 

15.3 

13.1 

11.5 

10.1 

9.1 

6.5 

5.7 

5.1 

4.G 

3.6 

3.0 

27 

11.5 

1.003 

1530.0 

14.2 

12.1 

10.0 

9.4 

8-5 

6.C 

5.3 

4.7 

4.2 

3.4 

2.8 

28 

11.1 

1.084 

1586.6 

13.2 

11  .f 

9.9 

8.8 

7-i 

5.C 

4.9 

4.4 

3.9 

3.1 

2.6 

29 

10.7 

1.170 

1643.3 

12.3 

10.5 

9.2 

8.2 

7.4 

5.2 

4.0 

4.1 

3.7 

2.9 

2.4 

30 

10.3 

1.257 

1700.0 

11.5 

9.8 

8.0 

7.G 

6.9 

4.! 

4.3 

as 

3.4 

2.7 

2.3 

31 

10.0 

L350 

1756.6 

10. 

9.2 

8.0 

7.1 

6.4 

4.C 

4.0 

3.6 

3.2 

2.5 

2.1 

32 

9.7 

1.443 

1813.3 

10.1 

8.6 

7.5 

67 

6.0 

4.3 

3.7 

3.4 

3.0 

2.4 

2.0 

33 

9.4 

1.546 

1870.0 

9.5 

8.2 

7.1 

6.P 

5.7 

4.( 

3.5 

3.1 

2.8 

2.2 

1.9 

34 

9.1 

1.650 

1926.6 

8.9 

7.6 

6.7 

5.9 

5.3 

3'.8 

3.8 

2.9 

2.6 

2.1 

1.7 

35 

8.8 

1.758 

1983.3 

8.4 

7.2 

6.; 

5.( 

5-0 

3.f 

3.1 

2.8 

2.5 

2.0 

1.6 

36 

8.6 

1.871 

2040.0 

8.0 

6.8 

6.0 

5.3 

4.8 

3.4 

3.0 

2.6 

2^4 

1.9 

1.6 

37 

8.4 

1.987 

2096.6 

7.5 

6.4 

5.6 

5.0 

4.5 

3.L 

2.8 

•2.5 

2.2 

I'.s 

1.5 

38 

8.1 

2.104 

2153.3 

7.1 

6.1 

5.3 

47 

4.3 

3.( 

2.6 

2.3 

2.1 

1.7 

1.4 

39 

7.9 

2.234 

2210.0 

6.8 

5.8 

5.1 

4.5 

4.0 

2.9 

2.5 

2.2 

2.0 

l.C 

1.3 

40 

7.7 

2.336 

2266.6 

6.4 

5.5 

4.8 

4.3 

3.8 

&u 

2.*4 

2.1 

1.9 

1.5 

1.2 

RESISTANCE   TO   CROSS  BREAKING  AND   SHEARING. 


45 


Fig.  84. 


12"  "Light"  Beam.    Weight  per  If.  =41.66  Ibs. 


Sectional  area =    12.5" 

Moment  of  inertia  1=  275.92 

Constant  K' =  229.94 

K' 


1 

CB 

d 

Distance  d  bet.  centres  of  beams  in  feet,  for 

ft^5 

3 
d 

.2 

1 

weight  in  Ibs.  per  sq.  foot  of  — 

CO    O 

£ 

d 

6 

d 

GO 

GO 

^ 

o! 

CD 

•£P 

CO 

CO* 

GO 

.0 

ja 

J3 

.0 

JO 

,0 

ft 

1 

1 

0 

.Q 

i 

1 

i 

g 

8 

o 

1 

| 

o 

I 

1 

6 

39.31 

0.047 

250.0 

7 

32.84 

0.063 

291.6 

8 

28.74 

0.083 

333.3 

24.0 

9 

25.54 

0.105 

375  0 

23.0 

18.9 

10 

22.98 

0.131 

416.6 

22.0 

18.3 

15.3 

11 

20.90 

0.158 

458.3 

23.0 

21.0 

19.0 

15.2 

12.6 

12 

19.16 

0.189 

500.0 

22.0 

19.9 

17.7 

15.9 

12.7 

10.6 

13 

17.68 

0.222 

541.6 

10  4 

17.0 

15.1 

13.6 

10.9 

9.0 

14 

16.42 

0.258 

583.3 

16.7 

14.6 

13.0 

9.3 

7.8 

15 

15.32 

0.297 

625.0 

22.0 

20.0 

14.5 

12.7 

11.3 

iol2 

8.1 

6^7 

16 

14.37 

0.339 

666.6 

22.0 

19.9 

17.9 

12.8  11.2 

9.9 

8.9 

7.1 

5.9 

17 

13.52 

0.383 

708  3 

19.9 

17.7 

15.9 

11.3    9  9 

8.8 

7.9 

6.3 

5.3 

18 

12.77 

0.431 

750.0 

20.0 

17.7 

15.7 

14.1 

10.1 

8.8 

7.8 

7.1 

5.6 

4'.7 

19 

12.10 

0.481 

791.6 

21.0 

18.3 

15.9 

14.2 

12.7 

9.1 

7.9 

7.0 

6.3 

5.1 

4.2 

20 

11.48 

0.538 

833.3 

19.1 

16.4 

14.3 

12.7 

11.4 

82 

7.1 

6.3 

5.7 

4.5 

3.8 

21 

10.94 

0.592 

875.0 

17.3 

15.0 

13  0 

ll.( 

10.4 

7.4 

6  5 

5  7 

5  2 

4  1 

3.4 

22 

10.44 

0.652 

916.6 

15.8 

13^5 

111  8 

me 

9^5 

G!? 

5.9 

5.2 

4.7 

3.7 

3.1 

23 

9.99 

0.717 

958.3 

14.4 

12.5 

10.8 

9.7 

8.6 

6.2 

5.4 

4.8 

4..° 

3.4 

2.8 

24 

9.58 

0.786 

1000.0 

13.3 

11.4 

9.9 

8.8 

7.9 

5.7 

4.9 

4.4 

3.9 

3.1 

2.6 

25 

9.19 

0.855 

1041.6 

12.2 

10.5 

9.1 

8.2 

7.3 

5.2 

4.5 

4.0 

3  ( 

2.9 

2.4 

26 

8.84 

0.927 

1083.3 

11.3 

9.7 

8.5 

7.5 

6.8 

4.8 

4.2 

3.7 

3.4 

2.7 

2.2 

27 

8.51 

1.003 

1125.0 

10.6 

9.0 

7.8 

7.0 

6.3 

4.5 

3.9 

3.5 

3.1 

2.5 

2.1 

28 

8.21 

1.084 

1166.6 

9.7 

8.3 

6.5 

5.8 

4.1 

3-.C 

3.2 

2.9 

2.3 

1.9 

29 

7.92 

1.170 

1208.3 

9.1 

7.8 

6.8 

6.1 

5.4 

3.8 

3.4 

3.( 

2'.7 

2.1 

1.8 

30 

7.66 

1.257 

1250.0 

8.5 

7.2 

6.3 

5  ( 

5.1 

3.6 

3.1 

2.8 

2.f 

2.0 

1.7 

31 

7.41 

1.350 

1291.6 

7.9 

6.8 

5.9 

5^3 

4.8 

3.4 

2.9 

2,( 

2.3 

1.9 

1.5 

32 

7.18 

1.443 

1333.3 

7.4 

6.4 

5.6 

4.9 

4.4 

3.2 

2.8 

2.4 

2.2 

1.7 

1.4 

33 

6.96 

1.542 

1375.0 

7.C 

6.0 

5.2 

4.7 

4.2 

3.0 

2.6 

2.3 

2.1 

i.r 

1.4 

34 

6.75 

1.645 

1416.6 

6.6 

5.6 

4.9 

4.4 

3.9 

2.8 

2.4 

2.2 

2.0 

1.5 

1.3 

35 

6.57 

1.754 

1458.3 

6.2 

5.3 

4.7 

4.1 

3.7 

2.6 

2.3 

2.0 

1.8 

1.5 

1.2 

36 

6.38 

1.871 

1500.0 

5.9 

5.0 

4.4 

3.9 

3.5 

2.5 

2.2 

1.9 

1.7 

1.4 

1.1 

37 

6.21 

1.987 

1541.6 

5.5 

4.8 

4.2 

3.7 

3.3 

2.3 

2.0 

1.8 

1.6 

1.3 

1.1 

38 

6.05 

2.109 

1583.3 

5.3 

4.5 

3.9 

3.5 

3.1 

2.2 

1.9 

1.7 

l.f 

1.2 

1.0 

39 

5.89 

2.229 

1625.0 

5.0 

4.J 

3.7 

3.3 

3.0 

2.1 

1.8 

1.6 

1.4 

1.1 

1.0 

40 

5.74 

2.366 

1666.6 

4.7 

4.1 

3.5 

3.1 

2.8 

2.0 

1.7 

1.5 

1.3 

1.0 

0.9 

46 


RESISTANCE   TO   CROSS  BREAKING  AND   SHEARING. 


10.5' 


Fig.  85. 


10.5"  Seam.     Weight  per  If.  =  35  Ibs. 


Sectional  area =   10.5" 

Moment  of  inertia  I  =  179.44 

Constant  K' =170.903 

K' 


-2 

c 

Distance  d  bet.  centres  of  beams  in  feet,  for 

g;  , 

c 

d 

£ 

weight  in  Ibs.  per  sq.  foot  of  — 

E"£ 

c 

"7 

«2 

c5 
1 

c 

c5 
o 

09 

1 

K 

A 

.c 

<R 

J§ 

| 

| 

| 

1 

| 

0* 

& 

3 

Q 

s 

o 
Q 

£ 

I 

I 

o 

00 

I 

I 

§ 

I 

I 

1 

I 

1 

6 

2U8 

0.053 

2100 

2L41 

0.072 

2450 

23.2 

8 

21.3(5 

0.095 

280.0 

21.3 

17.8 

9 

18  OS 

0.120 

315.0 

23.4 

21.1 

17.0 

14.0 

10 

17.09 

0.149 

350.0 

21  3 

18  9 

17.0 

13.6 

11.4 

11 

15.53 

0.181 

385.0 

20.1 

17.6 

15.6 

14.1 

11.3 

9.3 

12 

14.21 

0.216 

420.0 

1(5.9 

14.8 

13.1 

11.8 

9.4 

7.9 

13 

13  14 

0.254 

455.0 

20,6 

202  144 

12.6 

11.1 

10.1 

8.1 

6  7 

14 

12.2(1 

0.295 

490.0 

21.7 

19.2 

17.4 

12.4 

10.9 

9.6 

8.7 

6.9 

5*.8 

15 

11.38 

0.340 

525.0 

21.9 

18.9 

17.0 

15.1 

10.8 

9.4 

8.4 

7.5 

6.0 

5.0 

16 

1068 

0.389 

560.0 

22.2 

19.0 

16.6 

14.9 

13.3 

9.5 

8.3 

7.4 

6.6 

5.3 

4.4 

17 

10.05 

0.439 

595.0 

19.7 

17.0 

14.7 

13.2 

11.8 

8.4 

7.3 

6.5 

5.9 

4.7 

3.9 

18 

9.49 

0.494 

630.0 

17.5 

15.0 

131 

11.7 

10.5 

7.6 

6.5 

5.8 

5.2 

4.2 

3.5 

19 

8.99 

0.553 

6650 

15.7 

13.0 

11.7 

10.5 

9.4 

6.7 

5.9 

5.2 

4.7 

3.7 

3.1 

2) 

8.54 

0.614 

700.0 

14.2 

12.2 

10.6 

9.4 

85 

6.1 

5.3 

4.7 

4.2 

3.4 

2.8 

21 

8.13 

0.681 

735.0 

12.9 

11.1 

9.6 

8.6 

7.7 

5.5 

4.8 

4.3 

3.8 

3.1 

2.5 

22 

7.75 

0.752 

770.0 

11.7 

10.0 

9.1 

7.8 

7.0 

5.0 

4.4 

3.9 

3.5 

2.8 

2.3 

2> 

7.43 

0.823 

805.0 

10.7 

9.2 

8.0 

7.2 

6.4 

4.6 

4.0 

3.5 

3.2 

2.5 

2.1 

24 

7.12 

0.903 

840.0 

9.8 

8.4 

7.4 

6.5 

5.9 

4.2 

3.7 

3.2 

2.9 

2.3 

1.9 

25 

6.83 

0.980 

875.0 

9.1 

7.8 

6.8 

6.0 

5.4 

3.9 

3.4 

3.0 

2.7 

2.1 

1.8 

2', 

657 

1.067 

910.0 

8.4 

7.2 

6.3 

5.6 

5.0 

3.6 

3.1 

2.8 

2.5 

20 

1.6 

27 

G.32 

1.154 

945.0 

7.8 

6.7 

5.8 

5.2 

4.6 

3.3 

2.9 

2.6 

2.3 

1.8 

1.5 

28 

6.10 

1.251 

•980.0 

7.2 

6.2 

5.4 

4.8 

4.3 

3.1 

2.7 

2.4 

2.1 

1.7 

1.4 

2!) 

5.89 

1.346 

1015.0 

6.7 

5.8 

5.0 

4.5 

4.0 

2.9 

2.5 

2.2 

2.0 

1.6 

1.3 

30 

5.69 

1.450 

1050.0 

6.3 

5.4 

4.7 

4.2 

3.7 

2.7 

2.3 

2.1 

1.8 

.5 

1.2 

31 

5.51 

1.556 

1085.0 

5.9 

5.1 

4.4 

3.9 

3.5 

2.5 

2.2 

1.9 

1.7 

'A 

1.1 

32 

5.31 

1.672 

1120.0 

5.5 

4.7 

4.1 

3.7 

3.3 

2.3 

2.0 

1.8 

1.6 

.3 

1.1 

33 

5.17 

1.783 

1  155.0 

5.2 

4.4 

3.9 

3.4 

3.1 

2.2 

1.9 

1.7 

1.5 

.2 

1.0 

34 

5.02 

1.906 

1190.0 

4.8 

4.2 

3.6 

3.2 

2.9 

2.1 

1.8 

1.6 

1.4 

!T 

35 

4.88 

2.033 

1225.0 

4.6 

4.0 

3.4 

3.1 

2.7 

1.9 

1.7 

1.5 

1.3 

.1 

36 

4.69 

2.143 

1200.0 

4.3 

3.7 

3.2 

2.8 

2.6 

1.8 

1.6 

1.4 

1.3 

1.0 

37 

4.61 

2.297 

1295.0 

4.1 

3.5 

3.1 

2.7 

2.4 

1.7 

1.5 

1.3 

1.2 

38 

4.50 

2.444 

1330.0 

3.9 

3.3 

2.9 

2.6 

2.3 

1.6 

1.4 

1.3 

1.1 

39 

4.38 

2589 

1365.0 

3.6 

3.2 

2.8 

2.5 

2.2 

1.6 

1.4 

1.2 

1.1 

40 

4.20 

2.711 

1400.0 

3.5 

3.0 

2.6 

2.3 

2.1 

1.5 

1.3 

1.1 

1.0 

RESISTANCE  TO   CROSS- BREAKING  AND   SHEARING. 


47 


Fig.  80. 


9/x  "  Heavy"  Beam.     Weight  per  If.  =  50  Ibs. 


Sectional  area =    15.0" 

Moment  of  inertia  /  =  188.55 

Constant /f —  209.50 

K' 


•§ 

03 

03 

Distance  d  bet.  centres  of  beams  in  feet,  for 

N 

o 

fl 

C 

03 

weight  in  Ibs.  per  sq.  foot  of  — 

3  o 

O3  O 

U 

"3 

£ 

J.S 

6 

6 

o 

il 

03 

a' 

« 

03 

g 

03 

OB 

w 

g 

JC 

-A 

,0 

.2 

Q 

ft 

CS 

o 

o 
Q 

I 

p 

,Q 

O 

So 

i 

8 

O 

1 

1 

8 

1 

6 

36.91 

0.065 

300.0 

7 

29.92 

0.084 

3500 

8 

26.1  8 

0.111 

400.0 

21.8 

9 

23.27 

0.141 

450.0 

20.  (i 

17.2 

10 

20.95 

0.174 

'    500.0 

23.2 

20.9 

16.7 

13.9 

11 

19.01 

0.211 

550.0 

21.6 

19.1 

17.3 

13.7 

11.5 

12 

17.45 

0.253 

600.0 

20.7 

18.1 

16.1 

14.5 

11.6 

9.6 

13 

16.11 

0297 

650.0 

17.C 

15.4 

13.8 

12.3 

9.9 

8.2 

14 

14.96 

0.345 

700.0 

21.3 

15.2 

13.3 

11.8 

10.C 

8.5 

7.1 

15 

13.96 

0.398 

750.0 

20.6 

18.0 

13.2 

11.6 

10.3 

9.3 

7.5 

6.2 

16 

13.09 

0.454 

800.0 

20.4 

19.5 

16.3 

ll.(j 

10.2 

9.0 

8.1 

6.5 

5.4 

17 

12.32 

0.515 

850.0 

20.7 

18.1 

16.1 

14.5 

10.3 

9.0 

8.0 

7.2 

5.7 

4.8 

18 

11.63 

0.580 

900.0 

21.5 

18.4 

16.1 

14.3 

12.9 

9.2 

8.0 

7.1 

6.4 

5.1 

4.3 

19 

11.02 

0.648 

950.0 

19.3 

16.5 

14.5 

12.8 

11.6 

8.2 

7.2 

6.4 

5.8 

4.6 

3.8 

20 

10.47 

0.722 

1000.0 

17.4 

14.9 

13.0 

11.4 

10.4 

7.4 

6.5 

5.8 

5.2 

4.1 

3.4 

21 

9.97 

0.799 

1050.0 

15.8 

13.5 

11.8 

10.5 

9.4 

6.8 

5.9 

5.2 

4.7 

3.4 

3.1 

22 

9.52 

0.883 

11000 

14.4 

12.3 

10.3 

9.6 

8.2 

6.1 

5.4 

4.8 

4.3 

3.4 

2.8 

23 

9.10 

0.968 

1150.0 

13.1 

11.3 

9.8 

8.7 

7.9 

5.6 

4.9 

4.3 

3.9 

3.1 

2.6 

24 

8.72 

1.002 

1200.0 

12.1 

10.3 

9.0 

8.0 

7.2 

5.1 

4.5 

4.0 

3.6 

2.9 

2.4 

25 

8.34 

1.152 

1250.0 

11.1 

9.5 

8.3 

7.4 

6.6 

4.7 

4.1 

3.7 

3.3 

2.G 

2.2 

26 

8.05 

1.258 

1300.0 

10.3 

8.8 

7.7 

6.8 

6.1 

4.4 

3".  8 

3.4 

3.0 

2.4 

2.0 

27 

7.75 

1.363 

1:550.0 

9.5 

8.0 

7.1 

6.3 

5.7 

4.1 

3.5 

3.1 

2.8 

2.2 

1.9 

28 

7.48 

1.477 

1400.0 

8.9 

7.6 

6.6 

5.9 

5.3 

3.8 

3.3 

2.9 

2.6 

2.1 

1.7 

29 

7.22 

1.593 

1450.0 

8.3 

7.2 

6.2 

5.5 

4.9 

3.5 

3.1 

2.7 

2.5 

1.9 

1.6 

30 

6.98 

1.718 

1500.0 

7.7 

6.6 

5.8 

5.1 

4.6 

3.3 

2.9 

2.5 

2.3 

1.8 

1.5 

31 

6.75 

1.846 

1550.0 

7.3 

6.2 

5.4 

4.8 

4.3 

3.1 

2.7 

2.4 

2.1 

1.7 

1.4 

32 

6.54 

1.982 

1GOO.O 

6.7 

5.8 

5.1 

4.4 

4.0 

2.9 

2.5 

2.2 

2.0 

1.6 

1.3 

33 

6.34 

2.119 

1650.0 

6.4 

5.6 

4.8 

•4.2 

3.8 

2*.7 

2.4 

2.1 

1.9 

1.5 

1.2 

34 

6.16 

2.265 

1700.0 

6.0 

5.1 

4.5 

4.0 

3.6 

2.5 

2.2 

2.0 

1.8 

1.4 

1.2 

35 

5.98 

2.416 

1750.0 

5.6 

4.8 

4.2 

3.7 

3.4 

2.4 

2.1 

1.8 

1.7 

1.3 

1.1 

36 

5.81 

2.577 

1800.0 

5.3 

4.6 

4.0 

3.5 

3.3 

2^3 

2^0 

1.7 

1.6 

1.2 

1.0 

37 

5.66 

2.742 

1850.0 

5.0 

4.3 

3.8 

3.3 

3.0 

2.1 

1.9 

1.6 

1.5 

1.2 

38 

5.51 

2.918 

1900.0 

4.8 

4.1 

3.6 

3.2 

2.9 

2.0 

1.8 

1.6 

1.4 

1.1 

39 

5.37 

3.098 

1950.0 

4.5 

3.9 

3.4 

3.0 

2.7 

1.9 

1.7 

15 

1.3 

1.1 

40 

5.27 

3.289 

2000.0 

4.3 

3.7 

3.2 

2.9 

2.6 

1.8 

1.6 

1.4 

1.3 

l.C 

43 


RESISTANCE   TO   CROSS  BREAKING  AND   SHEARING. 


9"  "  Medium"  Beam.     Weight  per  If.  =  30  Ibs. 


Sectional  aren =      9.0" 

Moment  of  inertia  I  =  111.32 

Constant  K' =  123.09 

Kf 


•§ 

oc 
p 

<D 

Distance  d  bet.  centres  of  beams  in  feet,  for 

o 

0 

1 

weight  in  Ibs.  per  sq.  foot  of— 

<n£ 

i 

ft 

.2 

~j 

G5  S 
.0 

O 

9 

Itf) 

J 

d 

g 

" 

J3 

| 

00 

JO 

s 

JO 

1 

n 

.0 

s 

6 

O 

ft 

9 

p 

o 

§ 

s 

8 

§ 

8 

1 

1 

1 

6 

20.00 

0.062 

1800 

22  0 

7 

17.07 

0.085 

210.0 

25.0 

20.  ( 

16.0 

8 

15.40 

0.111 

240.0 

21.0 

21.0 

in  n 

15.0 

12^0 

9 

13.74 

0.141 

270.0 

21.0 

19.0 

100  15.0 

11.0 

10.0 

10 

12.36 

0.174 

300.0 

17.0 

15.0 

13.0  12.0 

9.8 

8^2 

11 

11.24 

0.211 

330.0 

22.0 

20.0 

14.0 

12.0 

11.01  10.0 

8.1 

6.8 

12 

10.30 

0.252 

300.0 

21.0 

19.0 

17.0 

12.0 

10.0 

9.51    8.5 

6.8 

5.7 

13 

9.51 

0.297 

390.0 

2J.O 

18.0 

10.0 

14.0 

10.0 

9.1 

8.1!    7.3 

5.8 

4.8 

14 

8.83 

0.345 

420.0 

21.0 

18.0 

15.0 

14.0 

12.0 

9.0 

7.8 

7.0i    6.3 

5.0 

4.2 

15 

8.24 

0.398 

450.0 

18.0 

15.0 

13.0 

12.0 

10.0 

7.8 

6.8 

6.11    5.4 

4.3 

3.6 

16 

7.73 

0.455 

480.0 

10.0 

13.0 

12.0 

10.0 

9.6 

6.9 

6.0 

5.3;    4.8 

3.8 

3.2 

17 

7.21 

0.511 

510.0 

14.0 

12.0 

10.0 

9.4 

8.4 

6.0 

5.3 

4.7i    4.2 

3.2 

2.8 

18 

6.87 

0.580 

540-0 

12.0 

10.0 

9.5 

8.4 

7.6 

5.4 

4.7 

4.2^    3.8 

3.0 

2.5 

19 

6.51 

0.650 

570.0 

11.0 

9.7 

8.5 

7.0 

6.8 

4.8 

4.2 

3.8    3.4 

2.7 

2.2 

20 

6.18 

0.722 

000.0 

10.0 

8.8 

7.7 

6.8 

6.1 

4.4 

3.8 

3.4 

3.0 

2.4 

2.0 

21 

5.88 

0799 

630.0 

9.3 

8.0 

7.0 

6.2 

5.6 

4.0 

3.5 

3.1 

2.8 

2.2 

1.8 

22 

5.02 

0.884 

660.0 

8.5 

7.2 

6.3 

5.0 

5.1 

3.6 

3.1 

2.8 

2.5 

2.0 

1.7 

23 

5.37 

0.969 

6DO.O 

7.7 

6.6 

5.8 

5.1 

4.6 

3.3 

2.9 

2.5 

2.3 

1.8 

1.5 

24 

5.15 

1.065 

720.0 

7.1 

6.1 

5.3 

4.7 

4.2 

3.0 

.  2.0 

2.3 

2.1 

1.7 

1.4 

25 

4.94 

1.157 

750.0 

6.5 

5.6 

4.9 

4  3 

3.9 

2.8 

2.5 

2.1 

1.9 

1.5 

1.3 

26 

4.83 

1.277 

780.0 

6.1 

5.3 

4.6 

4J 

3.7 

2.6 

2.3 

2.0 

1.8 

1.4 

1.2 

27 

4.58 

1.365 

810.0 

5.6 

4.8 

4.2 

3.7 

3.3 

2.4 

2.1 

1.8 

1.6 

1.3 

1.1 

28 

4.41 

1.476 

840.0 

5.2 

4.5 

3.9 

3.5 

3.1 

2.2 

1.9 

1.7 

1.5 

1.2 

1.0 

29 

4.20 

1.593 

870.0 

4.8 

4.1 

3.6 

3.2 

2.9 

2.0 

1.8 

1.6 

1.4 

1.1 

30 

4.12 

1.718 

900.0 

4.5 

3.9 

3.4 

3.0 

2.7 

1.9 

1.7 

1.5 

1.3 

1.0 

31 

3.99 

1.846 

930.0 

4.2    3.0 

3.2 

2.8 

2.5 

1.8 

1.6 

1.4 

1.2 

1.0 

32 

3.80 

1.982 

900.0 

4.0    3.4 

3.0 

2.6 

2.4 

1.7 

1.5 

1.3 

1.2 

33 

3.74 

2.119 

9900 

3.7 

3.2 

2.8 

2.5 

2.2 

1.0 

1.4 

1.2 

1.1 

34 

3.03 

2.235 

1020.0 

3.5 

3.0 

2.6 

2.3 

2.1 

1.5 

1.3 

1.1 

1.0 

35 

3.53 

2.416 

1050.0 

3.3 

2.8 

2.5 

2.2 

2.0 

1.4 

1.2 

1.1 

1.0 

36 

3.43 

2.577 

1080.0 

3.1 

2.7 

2.3 

2.1 

1.9 

1.3 

1.1 

1.0 

37 

3.34 

2.742 

1110.0 

3.0 

2.5 

2.2 

2.0 

1.8 

1.2 

1.1 

1.0 

38 

3.25 

2.918 

1140.0 

28 

2.4 

2.1 

1.9 

1.7 

1.2 

1.0 

39 

3.17 

3.098 

1170.0 

2.7 

2.3 

2.0 

1.7 

1.6 

1.1 

1.0 

40 

3.09 

3.289 

1200.0 

2.5 

2.2 

1.9 

1.6 

1.5 

1.1 

RESISTANCE   TO   CROSS  BREAKING  AND   SHEARING. 


Fig.  88. 


9"  "UgU"  Beam.     Weight  per  If.  =  23.33  Ibs. 


Sectional  area =     7.0" 

Moment  of  inertia  /==   91.00 

Constant  Kr =101.2 

K' 


3 

CQ 

e 

CO 

o 

Distance  d  bet.  centres  of  beams  in  feet,  for 

1  • 

3 
.5 

JG 

o 

c 

rc 

,Q 

weight  in  pounds  per  sq.  foot  of  — 

CO   0) 

G 

fl 

J.S 

o 

d 

HP 

a! 

i 

«-; 

02 

1 

.0 

| 

,0 

1 

.a 

JD 

01 

ft 

6 

ft 

'8 

§ 

£ 

i 

8 

5 

g 

1 

g 

1 

i 

1 

g 

16.86 

0.002 

140.0 

22.0 

18.7 

7 

14.45 

0  035 

103.3 

i;>5  0 

79  () 

20.0 

.0.0 

13.7 

12.05 

0.111 

180.0 

25.0 

19.7 

17.5 

15.8 

[2.0 

10.5 

9 

11.24 

0.141 

21o'o 

17.8 

15.6 

13.8 

10.1 

8*.6 

10 

10.12 

0.175 

233.3 

22.0 

20.0 

14.4 

12.0 

11.2 

iai 

8.0 

6.7 

11 

9.20 

0.212 

250.0 

21.0 

18.7 

10.7 

11.9 

10.4 

9.2 

8.3 

0.7 

5.5 

12 

8.43 

0.253 

280.0 

23.0 

20.0 

17.5 

15.0 

14.0 

10.0 

8.7 

7.8 

7.0 

5.6 

4.6 

13 

7.78 

0.297 

303.3 

19.9 

17.9 

14.9 

13.4 

11.9 

8.5 

7.4 

0.0 

5.9 

4.8 

3.0 

14 

7.22 

0.345 

320.0 

17.1 

14.7 

12.8 

11.4 

10.3 

7.3 

0.4 

5.7 

5.1 

4.1 

3.4 

15 

6.74 

0.398 

350.0 

14.9 

12.9 

11.2 

10.0 

8.9 

6.4 

5.0 

5.0 

4.5 

3.0 

2.$ 

16 

6.31 

0.453 

373.3 

13.1 

11.2 

9.8 

8.7 

7.8 

5.0 

4.9 

4.3 

3.9 

3.1 

2.6 

17 

5.95 

0.510 

390.0 

11.6 

10.0 

8.7 

7.8 

7.0 

5.0 

4.3 

3.8 

•3.5 

2.8 

2.3 

18 

5.02 

0.579 

420.0 

10.4 

8.9 

7.8 

0.9 

0.2 

4.4 

3.9 

3.4 

3.1 

2.5 

2.0 

19 

5.32 

0.048 

443.3 

9.3 

8.0 

7.0 

0.2 

5.0 

4.0 

3.5 

3.1 

2.8 

2.2 

1.8 

20 

5.00 

0.721 

466.6 

8.4 

7.2 

03 

5.0 

5.0 

3.0 

3.1 

2.8 

2.5 

2.0 

1.0 

21 

4.81 

0.797 

490.0 

7.6 

G.5 

5.7 

5.1 

4.5 

3.2 

2.8 

2.5 

2.2 

1.8 

1.5 

22 

4.59 

0.879 

513.3 

0.9 

5.9 

5.2 

4.0 

4.1 

2.9 

2.0 

2.3 

2.1 

1.7 

1.3 

23 

4.40 

0.908 

536.6 

0.3 

5.5 

4.7 

4.2 

3.8 

2.7 

2.3 

2.1 

1.9 

1.5 

1.2 

24 

4.21 

1.000 

500.0 

5.8 

5.0 

4.3 

3.8 

3.5 

2.5 

2.1 

1.9 

1.7 

1.4 

1.1 

25 

4.04 

1.151 

583.3 

5.3 

4-G 

4.0 

3.5 

3.2 

2.3 

2.0 

1.7 

1.0 

1.2 

1.0 

26 

3.89 

1.254 

GOO.G 

4.9 

4.2 

3.7 

3.3 

2.9 

2.1 

1.8 

1.0 

1.4 

1.1 

27 

3.74 

1.359 

030.0 

4.0 

3.9 

3.4 

3.0 

2.7 

1.9 

1.7 

1.5 

1.3 

10 

28 

3.00 

1.400 

653.3 

4.2 

3.0 

3.2 

2.8 

2.5 

1.8 

1.0 

.4 

1.2 

1.0 

29 

3.48 

1.582 

676.6 

4.0 

3.4 

3.0 

2.0 

2.4 

1.7 

1.5 

.3 

1.1 

30 

3.37 

1.711 

700.0 

3.7 

3.2 

2.8 

2.5 

2  2 

1.0 

1.4 

1.2 

1.1 

31 

3.20 

1.837 

723.3 

3.5 

3.0 

2.0 

23 

2.1 

1.5 

1.3 

.1 

1.0 

32 

3.10 

1.968 

746.6 

3.2 

2.8 

2.4 

2.1 

1.9 

1.4 

12 

l.o 

33 

3.00 

2.104 

770.0 

3.0 

2.0 

2.3 

2.0 

1.8 

1.3 

1.1 

34 

2.97 

2.250 

793.3 

2.9 

2.4 

2.1 

1.9 

1.7 

1.2 

l.o 

35 

2.89 

2.399 

810.G 

2.7 

2.3 

2.C 

1.8 

1.0 

1.1 

36 

2.81 

2.505 

840.0 

2.0 

2.2 

1.9 

1.7 

1.5 

1.0 

37 

2.73 

2.723 

803.3 

2.4 

2.1 

1.8 

1.0 

1.4 

38 

2.06 

2.898 

880.6 

2.3 

2.0 

1.7 

1.5 

1.4 

39 

2.59 

3.070 

910.0 

9  9 

1  0 

i  r 

1  4 

1  '.} 

40 

2.52 

3.250 

2.1 

1.8 

1.5 

1.4 

1.2 

50 


RESISTANCE   TO    CROSS-BREAKING  AND   SHEARING. 


8"  Beam.     Weight  per  If.  -=  21.66  Ibs. 


Soctionnl  area =    6.5" 

Moment  of  inertia  I  =  05.09 

Constant,  K' =  82.49 

K' 


-2 
"tf 

=Q 

C 

o> 

Distance  d  bet.  centres  of  beams  in  feet,  for 

p. 

P.*-' 

c  c 

3 

"3 

c 

ri 

JB 

weight  in  pounds  per  pq.  foot  of— 

"c  c 
ffi  — 

c; 

c 
d 

09 

32 

e 
2 

tC 

1 

1 

1 

Ja 

J§ 

| 

5 

B 

_C 

i 

1 

w 

X>     ' 

cr 

s 

6 

& 

3 

i 

0 

55 

i 

§ 

1 

g 

i 

o 
o 

i 

1 

G 

13.74 

0.070 

130.0 

oo  9 

18  3 

Ti  ° 

7 

11.78 

0.095 

151.6 

21.' 

18.'"> 

10.8 

13.-; 

11.2 

g 

10.30 

0.124 

173.3 

>-  Y 

18.3 

1  0.0 

14.  3 

12.8 

10.3 

8.5 

9 

9.16 

0.158 

195.0 

.'().:> 

1  l.fi 

12.7 

11.3 

10.1 

8.1 

6.7 

10 

8.23 

0.198 

216.6 

J0.5 

1X.3 

10.4 

11.7 

10.2 

9.1 

8.2 

6.5 

5.4 

11 

7.49 

0.238 

238.3 

22.C 

19.4 

17.0 

15.1 

l:j.G 

9.7 

8.5 

7.5 

6.8 

5.4 

4.5 

12 

6.87 

0.284 

260.0 

19.0 

163 

L4.8 

12.7 

11.4 

8.1 

7.1 

6.3 

5.7 

4.5 

3.8 

13 

634 

0.335 

281.6 

16.2 

14.0 

12.1 

10.9 

0  7 

0.9 

6.0 

5.4 

4.8 

3.9 

3.2 

14 

5.89 

0.390 

303.3 

14.0 

12.0 

10.5 

9.:} 

8.4 

0.0 

5.2 

4.0 

4.2 

3.3 

2.8 

15 

6.49 

0447 

325.0 

12,2 

10.5 

9.1 

8,1 

7.3 

5.2 

4.5 

4.0 

3.6 

2.9 

2.4 

16 

5.15 

0.511 

346.6 

10.7 

9.1 

8.0 

7,1 

6.4 

4.5 

4.0 

3.6 

3.2 

2.5 

2.1 

17 

4.85 

0.580 

368.3 

9.5 

».0 

7.1 

6.3 

5.7 

4.0 

3.7 

3.2 

2.8 

2.3 

1.9 

18 

4.58 

0.653 

390.0 

8.4 

7.2 

6.3 

5.6 

5.1 

3.8 

3.1 

2.7 

2.5 

2.0 

1.7 

19 

4.34 

0.731 

411.6 

7.6 

65 

5.7 

5.1 

4.5 

3.3 

29 

B5 

22 

1.8 

1.5 

20 

4.11 

0.810 

433.3 

6.8 

5.8 

5.1 

4.5 

4.1 

29 

2.5 

2.2 

2.0 

1.6 

1.3 

21 

3.92 

0.898 

455.0 

0.2 

5.3 

4.6 

4.1 

3.7 

2.6 

2.3 

2.0 

1.8 

1.4 

1.2 

22 

3.73 

0.989 

476.6 

5.6 

4.8 

4.2 

3.7 

3.4 

2.4 

2.1 

1.8 

1.6 

1.3 

1.1 

23 

358 

1.090 

498.3 

5.1 

4.4 

3.8 

3.4 

3.1 

2.2 

1.9 

1.7 

1.5 

1.2 

1.0 

24 

3.42 

1.192 

520.0 

4.7 

4.0 

3.5 

3.1 

2.8 

2.0 

1.7 

1.5 

'.4 

1.1 

25 

3.29 

1.300 

541.6 

4.3 

3.7 

3.2 

2.9 

2.6 

1.8 

1.5 

1.4 

1.3 

1.0 

26 

3.17 

1.417 

563.3 

4.0 

3.4 

3.0 

2.7 

2.4 

1.7 

1.4 

1.3 

1.2 

27 

3.05 

1.536 

585.0 

3.7 

3.2 

2,8 

2.5 

2.2 

1.6 

1.4 

1.2 

1.1 

28 

2.94 

1.602 

606.6 

3.5 

3.0 

2.6 

2.3 

2.1 

1.5 

1.3 

1.1 

1.0 

29 

2.84 

1.795 

628.3 

3.2 

2.8 

2.4 

2.1 

1.9 

1.4 

1.2 

1.0 

30 

2.73 

1.923 

650.0 

3.0 

2.6 

2.2 

2.0 

1.8 

1.3 

1.1 

| 

31 

2.66 

2.080 

671.6 

2.8 

2.4 

2.1 

1.9 

1.7 

1.2 

1.0 

2.56 

2.219 

693.3 

2.6 

2.2 

2.0 

1.7 

1.0 

1.1 

33 

2.49 

2.383 

715.0 

2.5 

2.1 

1.8 

1.6 

1.4 

1.0 

34 

2.42 

2.550 

736.6 

2.3 

2.0 

1.7 

1.5 

1.4 

35 

2.35 

2.722 

758.3 

2.2 

1.9 

1.6 

1.4 

1.3 

36 

2.29 

2.907 

780.0 

2.1 

1.8 

1.5 

1.4 

1.2 

37 

2.22 

3.084 

801.6 

2.0 

1.7 

1.5 

1.3 

12 

38 

2.17 

3.290 

823.3 

1.9 

1.6 

1.4 

1.2 

1.1 

39 

2.11 

3.484 

845.0 

1.8 

1.5 

1.3 

1.2 

1.0 

40 

2.06 

3.702 

866.6 

1.7 

1.4 

1.2 

1.1 

1.0 

] 

BESISTANCE  TO  CEOSS-BBEAZING  AND  SHEABING. 


51 


Fig.  90. 


7"  Seam.    Weight  per  If.  =  18.33  Ibs. 


Sectional  area ==   55" 

Moment  of  inertia  /  =  44.84 

Constant  #' =64.06 

Kf 


•e 

jj 

02 

Distance  d  bet.  centres  of  beams  in  feet,  for 

O 

p. 

1 

A 
o 

m 

.0 

weight  in  Ibs.  per  sq.  foot  of  — 

13  O 

£2 

.s 

""" 

OD  CD 

5 

c5 

0 

d 

JbK) 

03 

so 

, 

| 

t 

| 

.JO 

i 

i 

X 

s 

OS 
O 

1 

i 

| 

i 

O 
05 

3 

0 

1 

1 

1 

§ 

§ 

6 

10.67 

0.080 

110.0 

25.4 

22  2 

H).7 

17.7 

14.2 

11  8 

7 

9.15 

0.109 

18.6 

16.3 

14/> 

13,0 

10.5 

8'.7 

8 

8.00 

0.143 

14&6 

25.'6 

22.2 

20.  (! 

14.2 

12.5 

11.1 

10.0 

8.0 

6.6 

9 

7.11 

0.181 

165.0 

2*2.9 

19.7 

17.5 

16.8 

11.2 

9.8 

8.7 

7.9 

6.3 

5.2 

10 

6.40 

0.224 

183.3 

2L3 

18.2 

16.0 

14.2 

12.8 

9.1 

8.0 

7.1 

6.4 

5.1 

4.2 

11 

5.82 

0.272 

201.6 

17.6 

15.3 

13.2 

11.7 

10.5 

7.5 

6.6 

5.8 

5.2 

4.2 

3.5 

12 

5.33 

0.325 

220.0 

14.8 

12.6 

11.1 

9.8 

8.8 

6.3 

5.5 

4.9 

4.4 

3.5 

2.9 

13 

4.92 

0.382 

238.3 

12.6 

10.9 

9.4 

8.3 

7.5 

5.4 

4.7 

4.1 

3.7 

3.0 

2.5 

14 

4.56 

0.444 

256.6 

10.8 

9.3 

8.1 

7.2 

6.5 

4.6 

4.0 

3.6 

3.2 

2.6 

2.1 

15 

4.27 

0.513 

275.0 

9.4 

8.2 

7.1 

6.3 

5.7 

4.0 

35 

3.1 

2.8 

2.2 

1.8 

16 

3.99 

0.585 

293.3 

8.3 

7.1 

6.2 

5.5 

4.9 

3.5 

3.1 

2.7 

2.4 

1.9 

1.6 

17 

3.76 

0.665 

311.6 

7.3 

6.5 

5.5 

4.9 

4.4 

3.1 

2.7 

2.3 

2.1 

1.7 

1.4 

18 

3.55 

0.749 

330.0 

6.5 

5.6 

4.9 

4.3 

3.9 

2.8 

24 

2.1 

1.9 

1.5 

1.3 

19 

3.37 

0.840 

348.3 

5.9 

5.1 

4.4 

3.9 

3.5 

2.5 

2.2 

1.9 

1.7 

1.4 

1.1 

20 

3.20 

0.936 

366.6 

5.3 

4.5 

4.0 

3.5 

3.2 

2.2 

2.0 

1.7 

1.6 

1.2 

1.0 

21 

3.05 

1.038 

385.0 

4.8 

4.1 

3.6 

3.2 

2.9 

2.0 

1.8 

1.6 

1.4 

1.1 

22 

2.91 

1.146 

403.3 

4.4 

3.7 

3.3 

2.9 

2.6 

1.8 

1.6 

1.4 

1.3 

1.0 

23 

2.78 

1.257 

421.6 

4.0 

3.4 

3.0 

2.7 

2.4 

1.7 

1.5 

1.3 

1.2 

24 

2.66 

1.381 

440.0 

3.6 

3.1 

2.7 

2.4 

2.2 

1.6 

1.3 

1.2 

1.1 

25 

2.56 

1.504 

458.3 

3.4 

2.9 

2.5 

2.2 

2.0 

1.5 

1.2 

1.1 

1.0 

26 

2.45 

1.630 

476.6 

3.1 

2.6 

2.3 

2.0 

18 

1.4 

1.1 

27 

2.37 

1.775 

495.0 

2.9 

2.5 

2.1 

1.9 

1.7 

1.3 

1.0 

28 

2.27 

1.871 

513.3 

2.7 

2.3 

2.0 

1.8 

1.6 

1.2 

29 

2.20 

2.075 

531.6 

2.5 

2.1 

1.8 

1.7 

1.5 

11 

30 

2.12 

2.229 

550.0 

2.3 

2.0 

1.7 

1.5 

1.4 

1.0 

52 


RESISTANCE  TO   CROSS- BREAKING  AND  SHEARING. 


6"  Seam.     Weight  per  If.  =  13.33  Ibs. 


I 0.28" 

^pfc 


Sectional  area =.   4.0" 

Moment  of  inertia  I  =  22.5 

Constant  K' =  37.64 

K' 
W . 


-£ 

K 

or: 

0) 

Distance  d  bet.  centres  of  beams  in  feet,  for 

a  . 

1 

0 

00* 

weight  in  Ibs.  per  sq.  foot  of— 

§*« 

d 

.5 

s 

°?c£ 

£ 

d 

d 

"®.S 

05 

6 
o 

a 

n 

J 

j§ 

« 

03 

jQ 

£> 

JO 

2 

J 

J 

« 

s 

Bi 

o 

o 
Q 

1 

i 

i 

i 

o 

Ci 

8 

o 

o 

CD 

§ 

1 

1 

1 

6 

6.27 

0  094 

80.0 

23.2 

20.9 

14.9 

13.0 

11.6 

10.4 

8.3 

6.9 

7 

5.37 

o!l28 

93.3 

19.1 

17.3 

L5.3 

lo'.o 

9.5 

8.4 

7.6 

6.1 

5.1 

8 

0'.168 

106^6 

19.5 

16.8 

14.6 

13.0 

11.7 

8.5 

6.5 

5.8 

4.7 

3.9 

9 

4J8 

0.213 

120.0 

15.4 

13.4 

11.6 

10.4 

9.2 

6.6 

sis 

5.1 

4.G 

3.7 

3.1 

10 

3.75 

0.263 

133.3 

12.5 

10.7 

9.3 

8.3 

7.5 

5.3 

4.7 

4.1 

3.7 

3.0 

2.5 

11 

3.42 

0.320 

146.6 

10.3 

9.0 

7.7 

6.9 

6.2 

4.4 

3.8 

3.4 

3.1 

2.4 

2.0 

12 

3.13 

0.382 

160.0 

8.6 

7.0 

6.5 

5.7 

5.2 

3.7 

3.2 

2.9 

2.6 

2.0 

1.7 

13 

2.89 

0.450 

173.3 

7.4 

6.4 

5.5 

4.9 

4.4 

3.1 

2.7 

2.4 

2.2 

1.7 

1.4 

14 

2.68 

0.524 

186.6 

6.3 

5.4 

4.7 

4.2 

3.8 

2.7 

2.3 

2.1 

1.9 

1.5 

1.2 

15 

2.51 

0.607 

200.0 

5.5 

4.8 

4.2 

3.7 

3.3 

2.3 

2.1 

1.8 

1.6 

1.3 

1.1 

16 

2.34 

0.689 

213.3 

4.8 

41 

3.6 

3.2 

2.9 

2.0 

1.8 

1.6 

1A 

1.1 

17 

2.21 

0.786 

226.6 

4.3 

3.7 

3.2 

2.8 

2.5 

1.8 

1.6 

1.4 

1.8 

18 

2.09 

0.888 

240.0 

3.8 

3.3 

2.9 

2.5 

2.3 

1.6 

1.4 

1.2 

1.1 

19 

1.98 

0.995 

253.3 

3.4 

3.0 

2.6 

2  3 

2.1 

1.4 

1.3 

1.1 

20 

1.88 

1.110 

266.6 

3.1 

2.7 

2.3 

2.1 

1.8 

1.3 

1.1 

21 

1.79 

1.231 

280.0 

2.8 

2.4 

2.1 

1.9 

1.7 

1.2 

1.0 

22 

1.70 

1.350 

293.3 

2.5 

2.2 

1.9 

1.7 

1.5 

1.1 

23 

1.63 

1.493 

306.6 

2.3 

2.0 

1.7 

1.5 

A 

1.0 

24 

1.56 

1.641 

320.0 

2.1 

1.8 

1.6 

1.4 

.3 

25 

1.50 

1.787 

333.3 

2.0 

1.7 

1.5 

1.3 

.2 

26 

1.44 

1.950 

346.6 

1.8 

1.5 

1.3 

1.2 

.1 

27 

1.39 

2.129 

360.0 

1.7 

1.4 

1.2 

1.1 

28 

1.33 

2.286 

373.3 

1.5 

1.3 

1.1 

29 

1.29 

2.489 

386.6 

1.4 

1.2 

1.0 

30 

1.25 

2.698 

400.0 

1.3 

1.1 

RESISTANCE  TO   CROSS-BREAKING  AND   SHEARING. 


53 


CAST-IRON  BEAMS. 

Factor  of  rupture  0  for  cast-iron  beams  of  various  sections. 
The  factor  C  is  based  on  practical  experiments  by  Hodgkinson- 
Its  value  alters  with  the  different  proportions  of  the  cross-sections 
of  beam. 

Beam  supported  at  the  ends  ;  load  concentrated  at  the  center. 

Reference. 

0  =  Factor  of  rupture. 
W=  Breaking  weight  in  Ibs. 
A  =  Sectional  area  of  beam  in  square  inches. 
I  =  Distance  between  supports  in  inches. 
h  =  Height  of  beam  in  inches. 


Dimensions  in  inches.    6  =  Thickness  of  web  at  center  is  the 
unit. 

Fig.  92.          0.32  =  0.726 


5.125  =  11.646 


0.44=6 


0.47  =  1.066 
10.52  =  1.186 


Fig.  93. 


2.27  =  5.156 
.  =  3.20        (7=27292 

1.74  =  5.86 
JJT 


5.125  =  17.086 


.0.26  =  0.866 


0.30  =  6 


. 55  =  1.736 

l.'78~="5?936 
=  2.73        (7=28513 


BESISTANCE  TO   CEOSS-BEEAKING  AND  SHEAEING. 

Fig.  2^.  1.07  =  3.346 

=0.946 

5.125  =  166 


2.10  =  6.566 
=  2.88        (7=30330 


.  95. 


1.6  =  4.216 
" 


5.125  =  13.486 


.  315  =  0.826 


0.38  =  6 


0.53  =  1.396 


Fig.  96. 


'4.16  =  10.946 
4.33         C=  35262 

2.33  =  8.756 


5.125=19.266 


0.31  =  1.166 


=  6 


=  2.486 


6.67  =  25.076 
6.23        (7=44176 


RESISTANCE  TO   CROSS-BREAKING   AND   SHEARING. 


55 


10 
r^ 

CD 
CO 


a 

(M 


JO  'O 


56 


EESISTANCE  TO   CROSS-BREAKING  AND  SHEARING. 


°-5  ^ 

o"* 

C  -<3j 
rn    ~ 


& 

S 


jo  'O 


8 


RESISTANCE  TO   CROSS-BREAKING   AND   SHEARING. 


57 


o 

CD 


r-i     W)        ^  T3 

2     «    >>£  fl 

Cf-     -FH     -ua     o    O> 

00 

^D 

0  S  '§1  ® 

f  1  IS! 

3 

-s 

a 

CX  .2     cj   a  K 

^ 

T—  1 

2  H  H  g^ 

O 

rH 

a  g 

5        .2 

CN  O  - 


1-1 -Is 

I        S     r&     O 


CD        CD     CQ 

T-\   "*   " 


•^ 
-»j 
e8 


| 


a 

3 


1  S 


S 


o 
CD 


O  M 


58  RESISTANCE  TO  CROSS- BREAKING  AND  SHEARING. 

Load  concentrated  at  center:    W=  — r-,  or  Kl  —  L  W. 

t 

Beam  fixed  at  one  end;  principal  flange  at  top. 
Load  equally  distributed:   TF=  -^-p  or  JT1  =  2 .1.  W. 

£  .   V 

Kl 

Load  concentrated  at  free  end:    W—  -7-7-,  or  Kl  =  4 .1.  W. 

.     4 .1 

[NOTE.— The  more  the  sectional  area  is  contained  in  coefficient  ITi,  the 
more  is  the  section  economical.] 

EXAMPLE. — Section  No.  34.  Load  equally  distributed;  beam 
supported  at  both  ends;  thickness  of  web  =  1  inch:  thickness 
of  flange  =  1}  inch;  height  =  10  inches;  width  of  flange  =  5.9 
inches.  Distance  between  supports  =  20  feet  =  240  inches. 

Kl         658 
W  =  -jy-  =  —^-  =  5.48  tons  capacity. 

T  Tfi 

The  moment  of  resistance  of  cross-section =••  -— r- 


RESISTANCE  TO   CROSS-BREAKING   AND  SHEARING. 


Number  of 
section. 

^3  0 

•2P-S 

|IB 

Sectional 
area  in 
square 
incnes. 

Coefficient 
K.I 

Fig.  102. 

1 

6 

5.0 

10.0 

238 

2 

6} 

5.2 

10.7 

280 

^      ft 

^       i 

3 

7 

5.5 

11.5 

322 

^       j 

JT 

4 

7} 

5.7 

12.2 

364 

1 

5 

8 

6.0 

13  0 

420 

N™**. 

_Z"]lpl»^iP^  _.$., 

6 

gi 

6.2 

13.7 

476 

Fig.  103. 

7 

9 

6.5 

14.5 

532 

\ 

lj. 

— 

8 

9} 

6.7 

15.2 

602 

9 

10 

6.9 

15.9 

672 

4c 

10 

10} 

7.1 

16.6 

742 

11 

11 

7.4 

17.4 

812 

> 

^^^i  _N/___ 

12 

11} 

7.6 

18  1 

882 

K""*"  -4 

13 

12 

79 

18.9 

966 

Fig.  104. 

t27 

14 

12} 

8.1 

19.6 

1050 

! 

* 

15 

13 

8.4 

20.4 

1134 

^ 

i 

16 

13} 

8.6 

21.1 

1232 

% 

\   .  ~ 

17 

14 

8.8 

21.8 

1316 

% 

i 

1  Q 

1/11 

5^^^4^~.   1  " 

lo 

14}  ; 

9.0 

22.5 

1428 

k-"--B  -i" 

19 

15 

9.3 

23.3 

1526 

Fig.  105. 

20 

15} 

9.5 

24.0 

1624 

:   *      I  F 

21 

16 

9.8 

24.8 

1750 

.:•    1  I 
/  !   \  -\ 

22 

16} 

10.0 

25.5 

1848 

\k 

I  ! 

23 

17 

10.3 

26.3 

1960 

1  ! 

24 

17} 

10.5 

27.0 

2086 

'_  •'..  J.'"  ;•/;•:,•;%•  * 

[ 

—  jg  ->;_ 

25 

18 

10.8 

27.8 

2212 

60 


RESISTANCE   TO   CROSS-BREAKING  AND   SHEARING. 


Number  of 
section. 

§.2 

Width  B 
of  lower 
flange  in 
inches. 

Sectional 
area  in 
square 
inches. 

Coefficient 
K.i 

Fig.  106. 

26 

6 

4.5 

10.4 

224 

i-  -£-> 

27 

6} 

4.6 

11.1 

266 

t      i 

28 

7 

4.8 

11.8 

322 

y- 

29 

7j 

5.0 

12.5 

364 

'-       \ 

„ 

P 

30 

8 

5.2 

13.2 

420 

31 

gl 

5.4 

13.9 

476 

Fig.  107. 

32 

°2 

9 

5.6 

14.7 

532 

\J"/ 

33 

9} 

5.7 

15.4 

588 

P 

--    -y 

34 

10 

5.9 

16.2 

658 

" 

35 

10$ 

6.1 

16.9 

728 

36 

11 

6.3 

17.6 

798 

| 

i 

1 

37 

11} 

6.5 

18.3 

882 

,-  k—  >^-  —  >; 

38 

12 

6.7 

19.1 

952 

Fig.  108. 

39 

12} 

6.9 

19.8 

1036 

'^ 

ir 

40 

13 

7.1 

20.6 

1134 

| 

i 

41 

13} 

7.3 

21.3 

1218 

rr 

, 

42 

14 

7.5 

22.1 

1316 

I   ' 

^ 

1      ! 

43 

14} 

7.7 

22.8 

1414 

E]l^ 

!< 

:.  -#-  >!  ?* 

44 

15 

7.9 

23.6 

1512 

Fig.  109. 

45 

15J 

8.0 

24.3 

1610 

•>i           \/gi 

i  IT 

46 

47 

16 
16J 

8.2 
8.4 

25.1 
25.8 

1722 
1834 

i 

48 

17 

8.6 

26.5 

1946 

1  * 

^   1 

1 

49 

1VJ 

8.8 

27.2 

2072 

sk 

KA 

I  Q 

9.0 

28.0 

2198 

,_..-S-  —  H- 

O\J 

J.O 

RESISTANCE  TO   CKOSS-BREAKING  AND  SHEAEING. 


61 


Number  of 
section. 

Height  H 
in  inches. 

*!t« 

Sectional 
area  in 
square 
inches. 

Coefficient 
J£i 

I 

ig.  110. 

51 

6 

4.2 

10.5 

224 

\ 

2'! 

1"           T 

52 

6} 

4.3 

11.4 

266 

1              \ 
%              i 

53 

7 

4.5 

12.3 

308 

"  f 

54 

7} 

4.6 

12.9 

364 

^              I 

55 

8 

4.7 

13.6 

406 

z^flltll 

^ 

1  A      O 

K  • 

B  > 

56 

8i 

4.8 

14.3 

462 

I 

fy.  111. 

57 

9 

5.0 

15.0 

532 

V 

7 

58 

91 

5.1 

15.7 

588 

m 

% 

1 

\ 

59 

10 

5.3 

16.5 

658 

1 

J£ 

60 

10} 

5.4 

17.2 

728 

1 

\ 

\ 

—r—^—r—,            ' 

61 

11 

5.6 

17.9 

798 

62 

11} 

5.7 

18.6 

868 

*— 

^--  •- 

63 

12 

5.9 

19.4 

952 

f 

7^.  112. 

64 

12} 

6.0 

20.1 

1036 

p"~* 

^ 

65 

13 

6.3 

20.9 

1120 

3T 

66 

13} 

6.4 

21.6 

1204 

M 
w 

! 

67 

14 

6.6 

22.4 

1302 

W 

1                ^ 

\          -*•* 

i  A  f\r\ 

HHf 

wSm^. 

68 

14} 

6.7 

23.1 

14UU 

<_  J 

J-  >| 

69 

15 

6.9 

23.8 

1498 

J 

V  113. 

70 

15} 

7.0 

24.5 

1610 

\j£j 

_^ 

i  f 

hf     Q 

oc    q 

i  >7ns 

1 

lo 

/  .4 

ZiO  .  O 

1  <UO 

1 

\ 

72 

16} 

7.3 

26.0 

1820 

i 

± 

73 

17 

7.5 

26.8 

1932 

i 

} 

1 

74 

17} 

7.7 

27.5 

2058 

4^ 

fer-L 

B-  —  »! 

75 

18 

7.9 

28.3 

2184 

62  RESISTANCE   TO   CROSS -BREAKING  AND   SHEARING; 


o   . 

^g 

c^c 

-so 

tt 

s  ° 

^  o 

31&I 

S«M     §1 

I'SgJ 

o^ 

£* 

'a> 

W.S 

1  *  %•- 

o 
o 

Fig.  114. 

I  frr? 

76 

6 

4.0 

12.0 

224 

i 

77 

7 

4.1 

13.1 

308 

i 

* 

78 

8 

4.2 

14.4 

406 

-F?>.  115. 

79 

9 

4.4 

15.7 

518 

VT 

~A~ 

1     i 

80 

10 

4.6 

17.1 

644 

* 

|                  I 

81 

11 

4.8 

18.6 

784 

"|P||        ~~]  I 

i«—  *-  -H 

82 

12 

5.0 

20.0 

938 

%,  116. 

1"        ^        "" 

i 

83 

13 

5.2 

21.4 

1106 

aJ, 

^            ' 

84 

14 

5.5 

22.9 

1288 

1 

TS2P7' 

<L  —  jB  —  >| 

85 

15 

5.7 

24.4 

1484 

Fig.  117. 

^'           ^T- 

86 

16 

5.9 

25.8 

1694 

1               1     1 

i  1L 

87 

17 

6.2 

27.3 

1918 

1          1  i 

i_ 

88 

18 

6.4 

28.8 

2156 

^---^""^ 

RESISTANCE   TO   CROSS-BREAKING   AND  SHEARING. 


63 


Number  of 
section. 

51 

II 

"o 

gj 

(33  ^  fl 

5  -^   rf^ 

Sectional 
area  in 
square 
incnes. 

Coefficient 
JZX 

Fig.  118. 

89 

6 

5.6 

12.9 

294 

W.  ^ 

90 

6} 

5.8 

13,8 

336 

I   U           I    ''I 

91 

7 

6.0 

14.7 

392 

B 

_ 

92 

ff 

6.2 

15.5 

448 

v 

93 

8 

6.4 

16.4 

518 

j0gH|BHB-i*'-.  • 

94 

ii 

6.6 

17.3 

588 

s'  \e.  .ft  _>| 

2 

Fig.V&l 

95 

9 

6.9 

18.3 

658 

£.  „// 

96 

9} 

7.1 

19.2 

742 

T 

97 

10 

7.4 

20.2 

826 

& 

98 

10} 

7.6 

21.1 

910 

99 

11 

7.9 

22.1 

1008 

i 

100 

Hi 

8.1 

23.0 

1106 

. 

101 

12 

8.4 

23.9 

1204 

Fig.  120. 

102 

8.6 

24.8 

1302 

w~  \~ 

103 

13 

8.9 

25.8 

1414 

^       7T 

m        JL 

104 

13} 

9.1 

26.7 

1526 

1          t   -'^-": 

I    ! 

105 

1  HA 

14 

9.4 

9r> 

27.7 

1652 

1  '7/2/1 

i&" 

lUb 

*?J 

.  b 

28.5 

1754: 

E-—  -J5-—  »r^- 

107 

15 

9.8 

29.4 

1890 

Fig.  121. 

\%'"                 \%? 

108 

15} 

10.0 

30.3 

2030 

ITT 

109 

16 

10.3 

31.3 

2156 

%  \ 

'    |     i 

110 

16} 

10.5 

32.2 

2296 

tj  *i 

111 

17 

10.8  ' 

33.2 

2436 

1     ! 

112 

171 

11.0 

34.1 

2590 

-i 

113 

18 

11.3 

35.0 

2730 

<____JB—  V; 

64 


RESISTANCE   TO   CROSS-BREAKING  AND   SHEARING. 


I 

\ 

ig.  122. 

2%r 

Number  of 
section. 

1 

X3  O  bJD  _e 

T3~  G^ 

Sectional 
a  r  e  a  i  n 
square 
inches. 

Coefficient 
jffl. 

114 
115 
116 

6 
6} 
7 

5.3 
5.4 
5.6 

13.6 
14.4 
15.3 

280 
336 
392 

I 

J5 

r 

117 

7} 

5.7 

16.1 

448 

M 

118 
119 
120 

8 
3} 
9 

5.9 
6.0 
6.2 

17.0 

17.8 
18.7 

518 
588 
658 

^^iltl^^s 
v 

Fig.  123. 

\ 

S-— 

-*   121 

9i 

6.4 

19.6 

742 

1 

122 

10 

6.6 

20.5 

814 

r  |l23 

10} 

6.8 

21.4 

910 

v 

;  124 

11 

7.0 

22.4 

994 

T&T4  l^H  i 

125 
126 
127 
128 

11} 
12 
12} 
13 

7.2 
7.4 
7.6 
7.8 

23.3 
24.2 
25.1 

26.1 

1092 
1190 
1288 
1400 

Fig.  124. 

\ 

A 

^ 

J 

r 

129 

13} 

8.0 

27.0 

1512 

H 

i 

1 

130 
131 
132 

14 
14} 
15 

8.2 
8.4 
8.6 

27.9 
28.8 
29.8 

1624 
1750 
1876 

i  ;!%* 

.  —  ~i 

5— 

Fig.  125. 
\5/?'     \5/oy 

133 

15} 

8.8 

30.7 

2002 

=  ' 
$ 

/a 

t-x 

134 

16 

9.0 

31.6 

2142 

1 

~ 

| 

| 

135 

16} 

9.2 

32.5 

2282 

1 

< 

^ 

k 

136 
137 
138 

17 

18 

9.4 
9.6 
9.8 

33.5 
34.4 
35.3 

2422 

2562 
2716 

B-  — 

•» 

EESISTANCE   TO   CROSS-BREAKING  AND   SHEARING.  65 


Jf 

I 


Fig.  127. 
W 


Fig  128. 


.  129. 


o  . 

II 

C  o 

Is 


139 


140 


141 


142 


143 


144 


145 


146 


147 


148 


149 


150 


151 


10 


11 


12 


13 


14 


15 


16 


17 


18 


5.0 


5.1 


5.3 


5.5 


5.7 


6.0 


6.3 


6.5 


6.8 


7.4 


7.7 


8.0 


15.0 


16.4 


18.0 


19  7 


21.4 


23.2 


25.0 


26.8 


28.6 


30.5 


32.3 


34.2 


36.0 


66 


RESISTANCE   TO   CROSS -BREAKING  AND   SHEARING. 


VH 

N  „• 

( 

^> 

2  d 

2 

2*00 

^  .5  2  cr 

d 

2 

jo^ 

^"o 

£Z  0  1^2 

J    CJ    ^J 

jj>jj 

E  "^ 

c 

o 

5 

WH.m 

J 

CC    *  "'" 

0 

Fig.  130. 

152 

6 

6.3 

16.2 

336 

\ 

8  A 

153 

6} 

6.5 

17.2 

406 

1    ' 

154 

7 

6.7 

18.3 

476 

155 

7} 

6.9 

19.3 

546 

^ts                i 

156 

8 

7.1 

20.4 

616 

T'/'Tl^ 

~^1 

'^~_lMi__ 

',)    v/ 

157 

8} 

7.3 

21.5 

700 

(<.__-.£._—  >i 

-F 

</.  131. 

158 

9 

7.5 

22.6 

784 

\  7^' 

/ 

159 

91 

7.7 

23.6 

882 

: 

A 

: 

JB: 

160 
161 

10 

8.0 
8.2 

24.7 

25.8 

980 
1078 

; 

162 

11 

8.4 

26.9 

1190 

""//'  —  '  '  '' 

•//^i  '"'.-'" 

l~~7~r]  w 

163 

11} 

8.6 

28.0 

1302 

!<-—  j 

5—  -->; 

164 

12 

8.9 

29.1 

1428 

fty.  132. 
\&'/ 

165 

12} 

9.1 

30.1 

1554 

P 

~A~ 

166 

13 

9.3 

31.2 

1680 

IT 

167 

13} 

9.5 

32.3 

1806 

P 

168 

14 

9.8 

33.5 

1960 

%, 

1»S>F 

169 

14} 

10.0 

34.6 

2100 

\<-—jB  —  >i 

170 

15 

10.3 

35.7 

2254 

g.  133. 

171 

15} 

10.5 

36.8 

2408 

1? 

it  - 

172 

16 

10.8 

38.0 

2562 

1 

1  i 

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173 

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RESISTANCE   TO    CEOSS-BEEAKING  AND   SHEARING.  67 


Number  of 
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AI. 

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177 

6 

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336 

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7 

6.1 

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RESISTANCE   TO   CROSS-BREAKING  AND   SHEARING. 


Number  of 
section. 

ta»5 

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Kl. 

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21.0 

392 

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^^     *i 

191 

7 

7.1 

23.0 

532 

2 

i 

192 

8 

7.4 

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714 

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193 

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27.6 

896 

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194 

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1120 

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195 

11 

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32.5 

1372 

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196 

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1638 

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197 

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201 

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17 

10.4 
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45.2 

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2954 
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202 

18 

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50.4 

3766 

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RESISTANCE   TO   CROSS-BREAKING   AND   SHEARING. 


Number  of 
section. 

^  03 

ill 

Sectional 
area  in 
square 
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Coefficient 
JO. 

^.  142. 

L5$ 

203 

6 

8.0 

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206 

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208 

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EESISTANCE  TO   CROSS- BREAKING  AND  SHEARING. 


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1.4 

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8 

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RESISTANCE  TO   CROSS-BREAKING  AND   SHEARING. 


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RESISTANCE   TO   CROSS-BREAKING  AND   SHEARING. 


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11 

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16 

4.4 

61.8 

5628 

3 

3,j^  —. 

' 

569 

18 

17 

4.9 

64.8 

6006 

i 

570 

18 

18 

5.3 

67.6 

6384 

94 


RESISTANCE   TO    CROSS-BREAKING  AND   SHEARING. 


Number  of 
section. 

tec 
"9 

•^^^  c 

•'•''gtg  C 

1-2  2  * 

0  C5  ~  £ 

Coefficient 
.121. 

Fig.  102. 

571 

6 

9 

2.3 

26.6 

504 

~2' 

E 

] 

i 

572 

6 

10 

2.7 

29.4 

574 

• 

! 

573 

6 

11 

3.0 

32.0 

630 

.  . 

2 

— 

-2" 

574 

6 

12 

3.3 

34.6 

700 

-1   1 

575 

6 

13 

3.7 

37.4 

756 

1   ' 
1  ^ 

A  C\  C\ 

OO/j 

Fig.  163. 

576 
577 

6 

15 

4  .0 
4.3 

40.  U 
42.6 

bZb 

882 

'"F 

P: 

* 

_.. 

... 

~ 

578 

6 

16 

4.7 

45.5 

952 

—  i 

4 

ir 

i 

579 

6 

17 

5.0 

48.0 

1008 

2 

p 

& 

580 

6 

18 

5.3 

50.6 

1064 

- 

JW 

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1 

i 
i 

581 

7 

9 

2.3 

28.6 

686 

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582 

7 

1  0 

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31.4 

770 

-%.  164. 

583 

/ 

7 

11 

3.0 

34.0 

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—  -, 

584 

7 

12 

3.4 

36.8 

938 

f  \ 

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t 

585 

7 

13 

3.8 

39.6 

1036 

f/ 

586 

7 

14 

4.1 

42.2 

1120 

2 

s 
^ 

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587 

7 

15 

4.5 

45.0 

1204 

1 

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588 

7 

16 

4.9 

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m 

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1 

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589 

7 

17 

5.2 

50.4 

1372 

-%.  165.  ' 

590 

i 

7 

18 

5.6 

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1456 

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' 

5  ' 

591 

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9 

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10 

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11 

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<< 

7 

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594 

8 

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3.3 

38.6 

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595 

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596 

8 

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44.2 

1414 

- 

s 

? 

.. 

597 

8 

15 

4.5 

47.0 

1526 

fr 

—  - 

^.. 

-- 

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RESISTANCE   TO   CROSS  BREAKING  AND    SHEARING. 


95 


mberof 
action. 

5j 

-H* 

5  1|| 

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a 

J£ 

si.S 

^  c«.= 

$  Cq5'£ 

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8 

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2 

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599 

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52.6 

1750 

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if 

! 

600 

8 

18 

5.7 

55.4 

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2" 

m 

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601 

9 

9 

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32.2 

1064 

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~ 

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603 

9 

11 

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37.8 

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g.  163. 

604 

9 

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3.3 

40.6 

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3.7 

43.4 

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606 

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p 

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612 

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11 

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624 

11 

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RESISTANCE   TO    CROSS-BREAKING  AND   SHEARING. 


Number  of 
section. 

^  S 

*|«« 

.?5I| 

Sectional 
area  in 
square 
inches. 

Coefficient 
£X 

Fig.  162. 

625 

11 

15 

4.3 

52.6 

2576 

g 

'•/. 

w 

3 

— 

"T 

626 

11 

16 

4.7 

55.4 

2758 

t 

627 

11 

17 

5.1 

58.2 

2954 

t 

2 

^ 

JL 

628 

11 

18 

5.6 

61.2 

3136 

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11. 

629 
630 

12 
12 

11 
12 

2.4 
2.9 

42.8 
45.8 

2086 
2296 

Fig.  163. 

631 

12 

13 

3.3 

48.1 

2506 

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o 

~7/  / 

%jjr 

_._ 

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-A 

632 

12 

14 

3.7 

51.4 

2716 

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2 

1 

633 

12 

15 

4.1 

54.2 

2940 

£ 

s 

634 

12 

16 

4.6 

57.2 

3150 

1 

\  % 

1 

635 

12 

17 

5.0 

60.0 

3360 

JmMM^m^ 

j 

m 

%%; 

-y- 

636 

12 

18 

5.4 

62.8 

3570 

!<  jg— 

—  > 

Fig.  164. 

637 

13 

11 

2.2 

44.4 

2338 

,-N| 

638 

13 

12 

2.7 

47.4 

2576 

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.  1 

ii 

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T 

639 

13 

13 

3.1 

50.2 

2814 

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fa, 

i 

640 
641 

13 
13 

14 
15 

3.5 
4.0 

53.0 
56.0 

3052 
3290 

w. 

y 

i 

i 

642 

13 

16 

4.4 

58.8 

3528 

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t\ 

2" 

m 

& 

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643 

13 

17 

4.9 

61.8 

3780 

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644 

13 

18 

5.3 

64.6 

4018 

i  J-i 

5  1 

645 

14 

11 

2.0 

46.0 

2604 

646 

14 

12 

2.5 

49.0 

2870 

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647 

14 

13 

2.9 

51.8 

3136 

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n 
i 

1 

i 

648 

14 

14 

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3402 

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i 

649 

14 

15 

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650 

14 

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651 

14 

17 

4.7 

63.4 

4208 

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— 

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1 

RESISTANCE   TO   CROSS-BREAKING  AND   SHEARING. 


Number  of 
section. 

j="o 

.Sf.S 
'5  ^ 

-Ifs 

•0  ~  G 
T3  ^  5="^ 

c'rt  *  J 

Coefficient 
Jp. 

Fig.  162. 

652 

14 

18 

5.1 

66.2 

4152 

f 

653 

15 

12 

2.3 

50.6 

3164 

l^M^I                     1 
W#':                     ' 

654 

15 

13 

2.7 

51.4 

3444 

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<2  ||p        -uL 

655 

15 

14 

3.2 

56.4 

3738 

'•%ss'                   1 

656 

15 

15 

3.6 

•59.2 

4032 

yOC'. 

Fig.  163. 

657 
658 

15 
15 

16 

17 

4.1 
4.5 

62.2 
65.0 

4296 
4606 

}<-  $--->! 

659 

15 

18 

4.9 

67.8 

4900 

T 

660 

16 

13 

2.5 

55.0 

3742 

sm       & 

661 

16 

14 

3.0 

58.0 

4074 

\WM                                     \ 

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662 

16 

15 

3.4 

60.8 

4396 

£ 

663 

16 

Q    Q 

CO     0 

4.71  Q 

Fig.  164. 

664 

16 

17 

O  .  <J 

4.3 

DO  .  O 

66.6 

tfc  /  io 

5026 

i 

^^~  -  K- 

665 

16 

18 

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69.6 

5348 

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666 

17 

13 

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4060 

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2 

m       & 

667 

17 

14 

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59.6 

4410 

fm        « 

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668 

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15 

3.2 

62.4 

4760 

mm%^^%^\/^~  3  it 

tm^mM/2/m- 

669 

17 

16 

3.7 

65.4 

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670 

17 

17 

4.1 

68.2 

5460 

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671 

17 

18 

4.6 

71.2 

5810 

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tr 

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675 

18 
18 

15 
16 

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676 

18 

17 

3.9 

69.8 

5080 

£.. 

>--B  —  H 

677 

18 

18 

4.4 

72.8 

6258 

RESISTANCE   TO   CROSS  BREAKING   AND    SHEARING. 


STRENGTH  OF  WOODEN  BEAMS. 

Capacity  W  in  Ibs.  of  American  white  and  yellow  pine  beams,  joists, 
&c.,  from  1"  x  1"  to  15  x  15  in. 


The  modulus  of  rupture  is  taken  at- 


-  =  1250  Ibs..  or  8  times  safety, 


K'  =  tabulated  coefficient,  to  be  divided  by 

I  =  distance  between  supports  in  inches,  or  length  of  beams  in  inches 
from  support  to  free  end  of  beam. 


a 

Coefficient 

11 

Height  in 

"o  c 

1 

2 

3 

4 

5 

6 

7 

1 

1666 

6666 

15000 

26666 

41666 

60000 

81666 

1_1^ 

2500 

10000 

22500 

39999 

62499 

90000 

122499 

2 

3333 

13333 

30000 

53333 

83333 

120000 

163333 

2*^ 

4166 

16666 

37500 

66666 

104166 

150000 

204166 

3 

5000 

19999 

45000 

80000 

124999 

180000 

244999 

31^ 

5833 

23333 

52700 

93333 

145833 

210000 

285833 

4  * 

6666 

26666 

60000 

106666 

166666 

240000 

326(566 

41^ 

7499 

29999 

67500 

119999 

187499 

270000 

367499 

5 

8333 

33333 

75000 

133333 

208333 

300000 

408333 

5% 

9166 

36666 

82500 

146666 

229166 

330000 

449166 

6 

10000 

39999 

90000 

159999 

249999 

360000 

489999 

6% 

10833 

43333 

.  97500 

173333 

270833 

390000 

530833 

'  7 

11666 

46666 

105000 

186666 

291666 

420000 

571666 

71^ 

12500 

49999 

112600 

199999 

312499 

450000 

612499 

8 

13333 

53333 

120000 

213333 

333333 

480000 

653333 

8  14 

14166 

566(56 

127500 

22(5666 

354166 

510000 

694166 

9'  " 

14998 

59999 

135000 

239999 

374999 

540000 

734999 

9% 

15831 

63333 

142500 

253333 

395833 

570000 

775833 

10 

16666 

66(566 

150000 

266666 

416666 

600000 

816666 

10/4 

17500 

69999 

157500 

279999 

437499 

630000 

857599 

11 

18333 

73333 

1(55000 

293333 

458333 

660000 

898533 

11*^ 

19166 

76666 

172500 

306666 

479166 

690000 

939366 

12  * 

20000 

79999 

180000 

319999 

499999 

720000 

979999 

12^4 

20833 

83333 

187500 

333333 

520833 

750000 

1020833 

13  " 

21666 

86066 

195000 

346666 

541666 

780000 

1061666 

1314 

22500 

89)99 

202500 

359999 

562499 

810000 

1102499 

14 

23333 

93333 

210000 

373333 

583333 

840000 

1143333 

14K 

24166 

96(566 

217500 

386666 

604166 

870900 

1184166 

15  * 

25000 

99099 

225000 

399999 

624999 

900000 

1224999 

RESISTANCE   TO   CEOSS- BREAKING   AND   SHEARING. 


BEAMS  SUPPORTED  AT  THE  ENDS. 

K' 
Load  equally  distributed,  W  =  —  or  Kf  =  I W.       I 

K' 

Load  concentrated  at  centre,   W==  —  or  K'  —  21W.      2 

21 


BEAMS  FIXED  AT  ONE  END. 

K' 

Load  equally  distributed,  W  =  —  or  Kf  —  UW.     3 

K' 

Load  concentrated  at  free  end,  W  =  —  or  K'  =  SI  W.     4 


inches. 


8 

9 

10 

11 

12 

13 

14 

15 

106666 

135000 

166666 

201757 

240000 

281666 

326666 

375000 

159999 

202500 

249999 

302636 

360000 

422499 

489999 

562500 

213333 

270000 

333333 

403515 

480000 

563333 

653333 

750000 

266666 

337500 

416666 

504393 

600000 

704166 

816666 

937500 

319999 

405000 

499999 

605272 

720000 

844999 

979999 

1125000 

373333 

472500 

583333 

706151 

840000 

985833 

1143333 

1312500 

426666 

540000 

666666 

807030 

960000 

1126666 

1306666 

1500000 

479999 

607500 

749999 

907908 

1080000 

1267499 

1469999 

1687500 

533333 

675000 

833333 

1008787 

1200000 

1408333 

1633333 

1875000 

586666 

742500 

916666 

1109666 

1320000 

1549166 

1796666 

2062500 

639999 

810000 

999999 

1210545 

1440000 

1689999 

1959999 

2250000 

693333 

877500 

1083333 

1311423 

1560000 

1830833 

2123333 

2437500 

746666 

945000 

1166666 

1412302 

1680000 

1971666 

2286666 

2625000 

799999 

1012500 

1249999 

1513181 

1800000 

2112499 

2449999 

2812500 

853333 

1080000 

1333333 

1614060 

1920000 

2253333 

2613333 

3000000 

906666 

1147500 

1416666 

1714938 

2040000 

2394166 

2776666 

3187500 

959999 

1215000 

1499999 

1815817 

2160000 

2534999 

2939999 

3375000 

1013333 

1232500 

1583333 

1916696 

2280000 

2675833 

3103333 

3562500 

1066666 

1350000 

1666666 

2017575 

2400000 

2816666 

3266666 

3750000 

1119999 

1417500 

1749999 

2118453 

2520000 

2957499 

3429999 

3937500 

1173333 

1485000 

1833333 

2219332 

2640000 

3098333 

3593333 

4125000 

1226666 

1552500 

1910666 

2320211 

2760000 

3239166 

3756660 

4312500 

1279999 

1620000 

1999999 

2421090 

2880000 

3379999 

3919999 

4500000 

1333333 

1687500 

2083333 

2521968 

3000000 

3520833 

4083333 

4(587500 

1386666 

1755000 

216(5666 

2622847 

3120000 

3661666 

4246666 

4875000 

1439999 

1822500 

2249999 

2723726 

3240000 

3802499 

4409999 

5062500 

1493333 

1890000 

2333333 

2824605 

3360000 

3943333 

4573333 

5250000 

1546666 

1957500 

2416666 

2925483 

3480000 

4084166 

4736(566 

5437500 

1599999 

2025000 

2499999 

3026362 

3600000 

4224999 

4899999 

5625000 

100 


PEESSURE  OF  SUPPORTS. 


PRESSURE  ON  SUPPORTS. 
REACTION  OF  SUPPORTS. 

For  a  continuous  beam,  horizontal  or  inclined.  Load  W, 
equally  distributed,  and  supports  equal  distance  apart.  Appli 
cable  to  trussed  beams,  rafters,  or  beams  supported  by  three  or 
more  supports. 

Reference.     (Fig.  166.) 
W/  =  Weight  of  load  per  unit  of  length  in  Ibs. 

L  =  Distance  between  supports  in  units  of  length. 
PtPi,  P-2,  =  Pressure  on  supports  in  Ibs.,  counting  frorn  end 

support  to  center  of  beam. 

Mt  Mlt  M2  =  Moments  of  rupture  over  supports. 
ra,  m11  m2  =  Moments  of  rupture  between  supports. 

I,  ll}  12  =  The  distance  from  a  support  to  section  where 
moments  m,  mlt  mz  occur. 

By  this  table  the  pressure  upon  any  support,  from  3  to  9  in 
number,  can  be  ascertained;  also  the  moments  of  rupture.  The 
table  is  used  in  calculating  the  strains  in  roof  trusses,  &c. 

Fig.  166. 


Reactions 
or  pressure. 

Number  of  Supports. 

3 

4 

5 

7 

9 

P 

I 

0.375  WtL 
1.25     W,L 

0.4  W,L 
1.1   W,L 

0.3929TF,£ 
1.1429  W,L 
0.9286  W,L 

0.3942  W,L 
1.1346  WtL 
0.9615  W,L 
1.0192  W,L 

0.3^43  W,L 
1.13401^^ 
0.9G29  WtL 
U)103W,L 
0.9948  WtL 

Ml 
M2 
M3 
Ml 

0.125  W,L2 

0.1   17^2 

0.1071  W,L2 
0.0714  WtL  2 

0.1058  WL  2 
0.0769  WtL  2 
0.0865  WtL  2 

0.1057  WtL  2 
0.0773  WtL  2 
0.0850  IF,  L  2 
0.0824^1/2 

PRESS  flUE   ON  SUPPORTS. 


101 


Reactions 
or  pressure. 

Number  of  Supports. 

3 

4 

5 

7 

9 

m 
mi 

W*2 

ms 

0.0703  WtL  2 

0.08     WtL'i 
0.025   W,Li 

0.0772^1,2 
0.0364^,1,2 

0.0777  IF,£  2 
0.0340^1,2 
0.0434:^1/2 

0.0777  W4L  2 
0.0339  ir,L  2 
0.04381^1-2 
0.0412  WtL  2 

I 

1 

0.375  L 

0.4    L 

0.5  Z, 

0.3928  £ 
0.535     L 

0.3942  ^ 
0.5288  L 
0.4903  £ 

0.3943  I/ 
0.5283  L 
0.4922  £ 
0.5025  L 

Reference.     (Figs.  167,  168,  and  169.) 
W,  TTj,  PT2  =  Load  in  Ibs. 

^,  ^1}  12  =  Dimensions  in  units  of  length. 
P,  P1}  P2  =  Pressure  on  supports  in  Ibs. 


Three  supports, 
unequal  distances 
apart. 


Fig.  168. 


Load  equally  distributed: 


One  support,  and 
fixed  at  one  end. 


102  COMPRESSIVE  STRAIN  AND  PRESSURE  ON  SUPPORTS. 

ig.  169. 


Load  concentrated  at  free  end: 


One  snpport, 
and  fixed  at 
one  end. 


COMPRESSIVE  STRAIN  AND  PRESSURE  ON  SUPPORTS. 

SLOPING  BEAMS,  RAFTERS,  &c. 

Load  W  equally  distributed. 

For  the  cross- breaking  strain,  the  rafter,  &c.,  is  to  be  treated 
as  a  horizontal  beam  of  the  length  I.  (See  Compound  Strains  in 
Beam,  &c.) 

Reference. 

C=  Compression  in  direction  of  beam. 
jy=  Horizontal  strain  acting  on  support. 
V=  Pressure  on  supports. 

Lower  end  supported  vertically  and  horizontally ;  upper  end 
resting  on  inclined  support : 


Fig.  170. 


(7  =  —-  sin  .  v 


W 

H=  -—  sin.-y  cos.-y 
2 


W 
x  =-^-  (cos.-y)2 


RESISTANCE   TO   CRUSE  ING. 

Upper  end  fized;  lower  end  supported  horizontally : 
Fig.  171. 


103 


w 

~2~ 


Upper  end  resting  against  a  vertical  surface ;  lower  end  sup- 
>orted  vertically  and  horizontally  : 


port 


rtically  and  horizontally 
Fig.  172. 


2  sin.'u 

W 
H=—-  cotg  v 


RESISTANCE  TO  CRUSHING. 

STRENGTH  OF  COLUMNS,  PILLARS,  AND  STRUTS. 

Reference. 

A  =  Area  of  cross-section  in  inches. 

C==  Coefficient,  depending  on  the  material. 

I  =  Least  moment  of  inertia  of  cross-section. 
W '=  Capacity  of  column,  pillar,  or  strut  in  Ibs. 

a  =  Coefficient,  depending  on  the  material  in  respect  to  flexure. 

c  =  Coefficient,  depending  on  the  material. 

h  =  The  least  dimension  across  the  section  in  inches. 

k  =  Factor  of  safety. 

I  =  Length  of  column,  &c.,  in  inches. 

r  =  Least  radius  of  gyration. 


104 


RESISTANCE   TO   CRUSHING. 


To  find  the  square  of  the  radius  of  gyration  (r2)  of  a  plane 
about  a  given  axis,  divide  the  least  moment  of  inertia  by  the 


sectional  area  of  the  plane ;  that  is,  r2  =  — . 

Values  of—  For  Malleable  Iron.  For  Cast  Iron. 

C=  36,000    Ibs.             80,000  Ibs. 

c=  36,000     "                  3,200    " 

a==  0.000333  0.0025 


For  Dry  Timber. 

7,200  Ibs.  ' 
3,000    " 
0.004 


The  factor  of  safety  k  should  be,  for  wrought  iron  =  6;  for  cast 
iron  =  8;  for  timber  =  10.     This  applies  to  moving  loads. 

Case  1. 

Rounded  or  hinged  at  both  ends,  as  per — 
Fig.  173. 

For  square,  rectangular,  or  circular  cross-section : 

ir=i._c^_ 


For  any  other  cross-section: 

=  ~l  7^" 

i+-5r 


Case  2. 

Fixed,  or  having  a  flat  base  at  one  end,  and  rounded  or  hinged 
at  the  other,  as  per — 
Fig.  174. 

For  square,  rectangular,  or  circular  cross-section : 

W  =  —  °A 

k  p 


For  any  other  cross-section: 
1  CA 


1  + 


16  J2 
9.c.r2 


RESISTANCE   TO   CRUSHING. 


105 


Case  3. 

Fixed,  or  having  flat  bases  at  both  ends,  as  per — 
Fig.  175. 

For  square,  rectangular,  or  circular  cross-section : 
1  CA 

l+aJ^- 

For  any  other  cross-section: 
1  OA 


1  + 


EXAMPLES. 

Case  1. 
Rounded  at  both  ends: 

What  is  the  capacity  of  a  u  rought-iron  strut  of  the  annexed 
figure  and  dimensions? 

I  =  10  feet  =  120  inches. 
A  =  4.68  inches. 


r__    0.9X  3.58+5.1x0.38 


36000  X  4.68 


1  + 


1202 


36000  X  0.689 


12 


168480 

57600 
~^~  124804" 


=  3.227 


"37322" 

The  same  as  above,  in  Case  3,  fixed  at  both  ends: 
36000  x  4.69  168480 


1  + 

168480 


36000  x  0.689 


1  + 


14400 
24804 


106 


RESISTANCE   TO   CRUSHING. 


For  the  annexed  figure  and  dimensions  ;    otherwise,  same  as 
above  : 

A  =  7  inches. 


Case  1. 


\2»:  T-    1  X  43+3  x  I3 


Rounded  at  both  ends : 
Fig.  177. 

1  = 


12 


=  5.( 


36000  x  7                ,     252000 
-T*-1W~  =  * 8 =  21'00°  lbs' 

.  ^   A.    ^  —  v  O 


X  _ 

^  ^   36000  X  0.8 

Same  as  above,  in  Case  3,  fixed  at  both  ends  : 
36000  x  7  252000 


i_i 

36000  X  0.8 


-  =  1-  -  ~-r—  =  42>000  lbs- 
1 .  o 


Fixed  ends  : 


Case  3. 


What  is  the  capacity  of  a  cast-iro7i  pillar  of  the  annexed  figure 
and  dimensions? 

1=  10  feet=  120  inches. 
A  =  11  inches. 


i  X  43  —  7  X  33 


=  26.9 


80000  X  11  880000 

~T90~         *  S~25"  =       ' 

1+0.0025  — r-f- 


RESISTANCE  TO   CRUSHING. 


107 


For  the  annexed  figure  and  dimensions;  otherwise,  same  as 
above. 

Fig.  179. 

'  V 

TP=J- 


A  —  28  inches. 
80000  X  28 


. 


1  +  0.0025— J- 

_2240000_  =  179)200U)3. 
1.5625 


For  the  annexed  figure  and  dimensions;    otherwise,  same  as 
above. 

Fig.  180. 

A  =  22  inches. 


80000  X  22 


1202 
1  +  0.0025- 


1760000 
1.5625 


=  140,800  Ibs. 


To  find  the  capacity  of  a  Column,  Pillar,  or  Strut  of  any 
cross-section  by  the  following  Table : 

Find  how  many  times  the  least  dimension  h  across  the  section 

is  contained  in  the  length  I  of  column,  &c. — that  is,  — then 

multiply  the  corresponding  number  on  the  same  horizontal  line, 
under  K" ',  by  the  sectional  area  of  cross-section.     This  gives  the 
capacity  in  tons  of  2,000  Ibs. 
Let  I  =  Length  of  column,  &c. 

h  =  Least  dimension  of  cross-section. 

K"  =  Capacity  in  tons  of  one  square  inch  of  cross-section,  to 
be  multiplied  by  sectional  area  of  desired  cross- 
section. 

Various  sections  for  which  this  table  is  applicable: 
Fig.  181.  Fig.  182. 


108  RESISTANCE   TO   CEUSHING. 

Fig.  183.  Fig.  184. 


Fig.  186. 


Fig.  185. 


Fig.  187. 


Fig.  188. 


[NOTE. — This  table  is  strictly  correct,  only  for  columns,  Ac.,  with  circular 
or  rectangular  cross-section.  As  the  error  is  small,  it  may  be  used  for 
any  cross-section.] 

Example  explanatory  of  the  following  table. 

What  is  the  capacity  of  a  cast-iron  column  10  feet  =  120  inches 
long,  fixed  at  both  ends,  and  of  the  annexed  cross-section  and 
dimensions? 


Fig.  189. 


7  19() 

-f-  =  -=-  =  40  K"  for  40=1.000  tons. 
h  3 

Area=  6  inches. 

W=.  6  X  1  =  6  tons,  8  times  safety. 


BESISTANCE   TO   CRUSHING. 

Column,  &c.,  fixed  at  both  ends. 


109 


Cast  Iron—  eight  times  safety. 

Wrought  Iron  —  six  times  safety. 

I 
h 

K" 

I 
h 

K" 

I 

h 

K" 

I 
h 

K" 

I 

~h 

K" 

I 
£ 

K" 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

1 

4.987 

25 

1.951 

49 

0.714 

1 

2.999 

25 

2.487 

49 

1.674 

o 

4.950 

26 

1.858 

50 

0.689 

2 

2.996 

26 

2.452 

50 

1.644 

3 

4.890 

27 

1.771 

51 

0.666 

3 

2.991 

27 

2.418 

51 

1.615 

4 

4.807 

28 

1.689 

52 

0.644 

4 

2.984 

28 

2.383 

52 

1.585 

5 

4.705 

29 

1.611 

53 

0.623 

5 

2.975 

29 

2.348 

53 

1.557 

6  |     4.587 

30 

1.538 

54 

0.603 

6 

2.964 

30 

2.313 

54 

1.529 

7  |     4.450 

31 

1.469 

55 

0.584 

7 

2.953 

31 

2.277 

55 

1.501 

8  I     4.310 

32 

1.404 

56 

0.565 

8 

2.938 

32 

2.242 

56 

1.474 

9 
10 

4.158 
4.000 

33 

34 

1.343 

1.285 

57 
58 

0.548 
0.531 

9 
10 

2.921 
2.905 

33 

34 

2.206 
2.172 

57 

58 

1.4-18 
1.422 

11 

3.838 

35 

1.230 

59 

0.515 

11 

2.885 

35 

2.136 

59 

1  396 

12 

3.676 

36 

1.179 

60 

0.500 

12 

2.863 

36 

2.101 

60 

1.371 

13  !     3.514 

37 

1.130 

61 

0.485 

13 

2.841 

37 

2.067 

01 

1.347 

14  i     3.355 

38 

1.084 

02 

0.471 

14 

2.817 

38 

2.032 

62 

1.323 

15  I     3.200 

39 

1.041 

63 

0.457 

15 

2.792 

39 

1.998 

63 

1.299 

If.       3.048 

40 

1.000 

64 

0.445 

16 

2.766 

40 

1.963 

64 

1.276 

17 

2.902 

41 

0.961 

65 

0.432 

17 

2.738 

41 

1.930 

65 

1.253 

18 

2.762 

42 

0.924 

66 

0.420 

18 

2.711 

42 

1.896 

66 

1.228 

19 

2.628 

43 

0.889 

67 

0.409 

19 

2.680 

43 

1.863 

67 

1.209 

20 

2.500 

44 

0.856 

68 

0.398 

20 

2.650 

44 

1.831 

68 

1.187 

21 

2378 

45 

0.824 

69 

0.387 

21 

2.619 

45 

1.798 

69 

1.167 

22 

2.252 

46 

0.794 

70 

0.377 

22 

2.586 

46 

1.767 

70 

1.146 

23 

2.152 

47 

0.766 

71 

0.367 

23 

2.554 

47 

1.735 

71 

1.126 

24 

2.049 

48 

0.739 

72 

0.358 

24 

2.520 

48 

1.704 

72 

1.107 

110 


KESISTANCE   TO    CRUSHING. 


Strength  of  Columns,  Pillars,  or  Struts,  of  seasoned  wood,  round  or 

square  section. 

Fixed  at  both  ends.     All  dimensions  in  inches. 
Find  how  many  times  the  least  dimension  across  the  section  is 

TT 

contained  in  the  length  or  height  of  column,  &c.;  that  is,  — — - ; 

then  multiply  the  corresponding  figures  on  the  same  horizontal 
line  under  K"  by  the  sectional  area  of  cross-section.  This  gives 
the  capacity  of  column,  &c.,  in  tons  of  2,000  Ibs.,  10  times  safety. 

Reference. 

H=  Length  of  column,  &c. 
D  =  Least  dimension  of  cross-section. 

K"  =  Capacity  in  tons  of  one  square  inch  of  cross-section,  to 
be  multiplied  by  sectional  area  of  desired  cross-section. 

The  coefficient  C  for  white  and  yellow  pine  in  the  following 
table  is  taken  at  -h-f  {}£  =  600  Ibs.  for  safety : 

For  oak  at  S-J{J--  =  800  Ibs.  per  square  inch  for  safety. 
EXAMPLE. — What  is  the  capacity  of  a  pillar  of  oak,  section 
4x6  inches,  length  =  12  feet  =  144  inches  ? 


for  36  _.  o.064  x  4  X  6  =  1.536  tons. 


Capacity  Kf/  of  one  square  inch  in  tons  of  2,000  Ibs. 


White  and  Yellow  Pine. 

Oak. 

H 

~D   " 

j\" 

H 
~D   = 

K" 

H 

~D   ~ 

E» 

H 

~D   ~ 

K" 

1 

0.299 

26 

0.081 

1 

0.399 

26 

0.108 

2 

0.2:)5 

27 

0.076 

2 

0.394 

27 

0.102 

3 

0.289 

28 

0.072 

3 

0.386 

23 

0.096 

4 

0.282 

29 

0.068 

4 

0.376 

29 

0.091 

5 

0.272 

30 

0.065 

5 

0.363 

30 

0.086 

6 

0.262 

31 

0.061 

6 

0.349 

31 

0.082 

7 

0.251 

32 

0.058 

7 

0.334 

32 

0.078 

8 

0239 

33 

0.056 

8 

0,319 

33 

0.074 

9 

0.226 

34 

0.053 

9 

0.302 

34 

0.071 

10 

0.214 

35 

0.050 

10 

0.285 

35 

0.067 

11 

0.202 

36 

0.048 

11 

0239 

36 

0.064 

12 

0.190 

37 

0.046 

12 

0.254 

37 

0.061 

13 

0.179 

38 

0.044 

13 

0.238 

38 

0.059 

14 

0.168 

39 

0.042 

14 

0.224 

39 

0.056 

15 

0.158 

40 

0.040 

15 

0.210 

40 

0.054 

16 

0.148 

41 

0.038 

16 

0.197 

41 

0.051 

17 

0.139 

42 

0.037 

17 

0.185 

42 

0.049 

18 

0.130 

43 

0.035 

18 

0.174 

43 

0.047 

19 

0.123 

44 

0.034 

19 

0.163 

44 

0.045 

20 

0.115 

45 

0.033 

20 

0.154 

45 

0.044 

21 

0.108 

46 

0.031 

21 

0144 

46 

0.042 

22 

0.102 

47 

0.030 

22 

0.136 

47 

0040 

23 

0.096 

48 

0.029 

23 

0.123 

48 

0.039 

21 

0.030 

49 

0.088 

24 

0.121 

49 

0.037 

25 

0.085 

50 

0.027 

25 

0.114 

50 

0.036 

PARALLELOGRAM   OF   FORCES. 


Ill 


PARALLELOGRAM  OF  FORCES. 
COMPOSITION  AND  RESOLUTION  OF  FORCES. 

Reference. 

A,  B,  C  =  Forces,  cr  strains,  acting  on  a  single  point, 
•y,  i/,  =  angles. 


Fig.  190. 


A  = 


(7sin.  v, 
sin.  (v  +  v,)  ' 


_  Csin.v  . 

B  =  — - — ; — -,  when  v  =  v/y  A  =  B  — 


sin.  (v  +  ^/)  ' 


•  sec.  v; 


whenv+^<90°       C  = 
when  t> -f  uy  >  90°       /?_ 


cos.  (v  +  v,) 


=  90° 
=  C  cos.  v 
=  C  sin.  v 
(7  = 


112 


STRAINS   IN  FRAMES. 


STRAINS  IN  FRAMES. 

Reference. 

C  =  Compressive  strain  in  units  of  weight, 
T=  Tensile 
F=  Vertical 
H=  Horizontal        " 
W=  Load  in  units  of  weight. 
I  =  Dimensions  in  units  of  length. 
v  =  Angle  between  horizontal  and  inclined  member. 
For  01  oss-breaking  strain,  see  "Resistance  to  cross-breaking.' 


Fig.  193. 


W 


2  sin.  v 

W 
yx=  — — —  cotg.  v  =  H 


Fig.  194. 


C/=  H  =  ij  W  cotg.v  =  cross-breaking  strain  at  H. 

H/=  -j-  H '=  j^.— j-  TPrcotg.  v  =  tension  in  H/. 
I  I 

H  —  H,  •=  \\.  (  — — --  )  W  cotg.  v  =  compression  in 
F=  U  W.         \      I      J 


STRAINS    IN"    FRAMES. 


113 


Fig.  195. 


T7  TT  - 

V  =  H/  tang,  y  =  —  -  tang,  y 


=  compression. 

W.I 

cos.  y  1^, ,  cos.  y 

nressiou. 

//=  W.I. 


When  I  >  ^3  the  portion  t//  is  in  tension  =  V  —  W  = 
W   Ltang.y-l) 

\    Ijj  I 

When  I  <  Z3  the  portion  ly/  is  in  compression  =  IF—  F  = 


F/  = — — .  TF=  tension, 

v 


/*'/.  107. 


Ends  of  beams  built  into  wall  or  fixed- 
F=  -L  W 


V,  =  V-  W  =  (—,-±-}   W,  =  T,  (tension)  =  C,  (. 

V  V  ' 

pression.) 


Ill  STRAINS   IN   BOOM    DERRICKS. 


C=  ( —  J  — =  (compression)  =  T  (tension.) 

V       21,      J    sin.  v 

II  —  f~ —\  TFcotg.  v  =  (tension)  =  H,  (compression.) 

Ends  of  beams  not  built  into  wall  or  fixed: 


T//r=  v—  W=  (— /— )  W=  C/  (compression)  =  T, 
(tension.) 

0= = =  T  (tension.) 

sin.  v          I,  sin.  v 

H=  Fcotg.  v  = TFcotg.  v  =  (tension)  =  H/  (compression.) 


STRAINS  IN  BOOM  DERRICKS. 

Reference. 

C  =  Compression  in  boom. 

C/=  Compression  in  mast. 

T=  Tension  in  tackling. 

T,=  Tension  in  guy. 

t  =  Tension  in  runner  from  mast  head  to  weight. 

t/=  Tension  in  runner  from  boom  head  to  weight. 
W  =  Weight  or  load. 
H  =  Horizontal  strain. 

V  =  Vertical  strain. 
v,  vlt  v2  =  Angles.     (See  Figure.) 


Fifj.  198. 


STEAINS   IN   TRUSSES. 


115 


TTsin.  v1 

sin.  (v  +  Vj 

F"==  £/  cosin.  v1 

(7=  Fcosec.  v2 

T=  Fcosec.  v» 


TFsin.  v 
sin.  (v  +  vx 


*,= 


<?,=  PF 

2^=  Fcotg.  -y3  sec.  v4 


STRAINS  IN  TRUSSES. 
Zoac?  equally  distributed. 

Reference. 

W=  Load  equally  distributed  in  Ibs. 
I  =  Distance  between  abutments. 
v  =  Angle  between  horizontal  and  diagonal. 
0=  Compression  in  Ibs.,  (denoted  by  thick  lines.) 
T=  Tension  in  Ibs.,  (denoted  by  thin  lines.) 

2  Bays  =  4- 


Fig.  199. 


T-    • 

-*    ~   T6    -    • 


3  Bays  =  — 


.  200. 


116 


STEAINS   IN  TRUSSES. 


4  Bays 

Fig.  201. 


C-  W 

C2 


3(72 


cotg. 


Fig.  202. 


C  =  T  —  '6C2  cotg.  v 
(?!=  T1=  2(72  cotg.  t? 

W 


2rT2=  ^  cosec.  v 


m 


STRAINS  IN  TRUSSES.  ' 
I 


0=  T  = 


90 


2 
8C, 


2 

5  a 


cotg.  v 
cotg.  v 


a— 


2 

5(73 

65  "    "~T~ 

T3  =  —j—  cosec. 

^4  =  3^3 


TABLE  OF  CONSTANTS,  BASED  ON  FOREGOING  FORMULA. 
Load  equally  distributed. 

Table  of  constants  for  strains  in  respective  member  of  trusses, 
from  2  to  6  bays,  with  diagonals  inclined  from  5°  to  45° : 

Reference. 

W  =  Load  in  Ibs.,  equally  distributed  over  whole  length  of 
truss,  to  be  multiplied  by  constant  for  strain  in  re- 
pective  member. 

v  =  Angle  between  horizontal  and  diagonal. 
0=  Compression  in  Ibs.  in  respective  member. 
T=  Tension  in  Ibs.  in  respective  member. 
EXAMPLE. — Required,  the  strain  in  the  various  members  of  a 
truss  of  4  bays.     Length  =  40  feet ;  load  W  =  80,000  Ibs. ;  angle 
v  =  20°. 

Members.    Constants.         W.  Strains. 

C  =  T  =  1.372  x  80,000  =  109,760  Ibs. 
ci=  TI=  1.029  X  80,000  =  82,320 
<72=0.25  x  80,000=  20,000 
<73=  0.375  x  80,000  =  30,000 
T2=  0.365  x  80,000=  29,200 
T3=  1.095  x  80,000  =  87,600 

[NOTE.— When  the  trusses  are  inverted,  the  strains  change  in  kind,  but 
not  in  amount] 


118 


STRAINS   IK  TRUSSES. 


2  Bays  =  - 

Fig.  204. 


3  Bays  =  — 
Fig.  205. 


V 

O 

C'l 

T 

(7=  T 

Ci 

Tl 

5 

3.572 

0.625 

3.584 

3.810 

0.333 

3.820 

6 

2.972 

" 

2987 

3.170 

" 

3.186 

7 

2.544 

<« 

2.562 

2.713 

« 

2.733 

8 

2.225 

" 

2.244 

2.370 

M 

2.393 

9 

1.972 

" 

1.997 

2.103 

" 

2.130 

10 

1.772 

(C 

1800 

1.890 

M 

1.920 

11 

1.610 

" 

1.640 

l.MO 

" 

1.747 

12 

1.469 

" 

1.500 

1.570 

M 

1.603 

13 

1.353 

" 

1.390 

1.444 

" 

1.483 

14 

1.253 

'< 

1.290 

1.333 

« 

1.376 

15 

1.166 

1 

1.210 

1.243 

" 

1.286 

16 

1.087 

" 

1.134 

1.160 

(« 

1.210 

17 

1.022 

" 

1.070 

1.090 

M 

1.140 

18 

0.959 

" 

1.013 

l.<23 

" 

1.080 

19 

0.906 

H 

0.959 

0.970 

" 

1.023 

20 

0.859 

" 

0.912 

0.917 

" 

0.973 

21 

0.813 

" 

0.872 

0.866 

" 

0.930 

22 

0.778 

" 

0.834 

0.823 

« 

0.890 

23 

0.734 

» 

0.790 

0.783 

«: 

0.853 

24 

0.703 

«( 

0.765 

0.750 

« 

0.810 

25 

0.668 

" 

0.738 

0.713 

" 

0.786 

26 

0.641 

« 

0.712 

0.685 

« 

0.760 

27 

0.613 

" 

0.687 

0.653 

« 

0.730 

28 

0.587 

«( 

0.666 

0.626 

" 

0.701 

29 

0.562 

<C 

0.644 

0.600 

.( 

0.686 

30 

0.541 

(( 

0.625 

0.643 

M 

0.666 

31 

0.519 

« 

0.606 

0.555 

" 

0.646 

32 

0.500 

M 

0.591 

0.533 

M 

0.630 

33 

0.481 

it 

0.575 

0.513 

« 

0.613 

34 

0.463 

0.559 

0.493 

0.596 

35 

0.447 

« 

0.544 

0.476 

" 

0.580 

36 

0.431 

« 

0.531 

0.460 

« 

0.566 

37 

0.416 

" 

0.519 

0.444 

" 

0.553 

38 

0.400 

" 

0.506 

0.426 

M 

0.540 

39 

0.384 

<• 

0.497 

0.410 

«« 

0.530 

40 

0.372 

« 

0.487 

0.396 

" 

0.520 

41 

0.359 

« 

0.475 

0.385 

" 

0.506 

42 

0.347 

H 

0.466 

0.370 

« 

0.496 

43 

0.334 

« 

0.456 

0.357 

" 

0.486 

44 

0.322 

<' 

0.450 

0.343 

« 

0.480 

45 

0.312 

" 

0.444 

0.333 

" 

0.473 

STRAINS   IN    TRUSSES. 


119 


4  Bays  =  — 
Fig.  20(>. 


V 

C'=  T 

CV-5T! 

C2 

Cs 

T2 

T'3 

5 

5.720 

4290 

0.250 

0.375 

1.434 

4.032 

6 

4.700 

3.570  • 

" 

11 

1.200 

3.600 

7 

4.008 

3.051 

" 

" 

1.025 

3.075 

8 

3.500 

2.070 

" 

0.897 

2591 

9 

3.1G4 

2.373 

'* 

;; 

0.799 

2.397 

10 

2.8-^2 

2.124 

0.720 

2.160 

11 

2.508 

1.926 

•; 

0.655 

1.965 

1'2 

2.388 

1.791 

«• 

0.601 

1.803 

13 

2.164 

1623 

0.556 

1.608 

14 

2.000 

1.500 

• 

0.516 

1.548 

15 

1.804 

1.398 

0.482 

1.446 

1C 

1.7-10 

1.305 

0.454 

1.362 

17 

i.632 

1.224 

'• 

0.428 

1.284 

18 

1.532 

1.149 

0.405 

1.215 

19 

1.448 

1.086 

'• 

0.384 

1.152 

20 

1.372 

1.029 

" 

0.365 

1.095 

21 

1.300 

0.975 

" 

•' 

0.349 

1.047 

22 

1.23G 

0.927 

'; 

•' 

0.334 

1.002 

23 

1.172 

0.879 

" 

0.32) 

0.960 

24 

1.124 

0.843 

'; 

" 

0.306 

0.918 

25 

1.008 

0801 

'• 

•' 

0.295 

0.885 

26 

1.024 

0.708 

'* 

0.285 

0.855 

27 

0.080 

0.735 

' 

0.275 

0.825 

2-i 

0.940 

0.705 

'• 

" 

0.266 

0.798 

2y 

0.900 

0.675 

1 

0.258 

0.774 

30 

0.804 

0.048 

•' 

'• 

0.250 

0.750 

31 

0.823 

0.621 

«« 

0.243 

0.729 

32 

0.800 

O.COO 

•' 

• 

0  236 

0.708 

H,i 

0.708 

0.576 

0.230 

0.690 

34 

0.740 

0.655 

*• 

0.224 

0.672 

35 

0.720 

0.540 

" 

* 

0.218 

0.654 

36 

0.088 

0.516 

0.212 

0.636 

37 

O.G04 

0.498 

'• 

0.207 

0.621 

38 

0.640 

0.480 

" 

0.203 

0.009 

39 

0.016 

0.462 

'• 

0.199 

0.597 

40 

0.000 

0.450 

'; 

• 

0.195 

0.585 

41 

0.576 

0.432 

• 

0.190 

0.570 

42 

0.500 

0.420 

' 

0.186 

0.558 

43 

0.536 

0.402 

0.183 

0.549 

44 

0.520 

0  390 

•  ' 

0.180 

0.540 

45 

0.500 

0.375 

^ 

" 

0.177 

0.531 

STRAINS   IN   TRUSSES. 


V 

e-jr 

C'i  =  Ti 

C'2 

^3 

Tz 

n 

5 

6.858 

4.572 

0.200 

0.400 

2.294 

4.588 

6 

5.706 

3.804 

" 

•' 

1.912 

3.824 

7 

4.884 

3.256 

' 

1.640 

3.280 

8 

4.272 

2.848 

" 

1.436 

2.872 

9 

3.786 

2.524 

«. 

1.278 

2.556 

10 

3.402 

2.268 

i 

1.152 

2.304 

11 

3.084 

2.056 

1.048 

2.0')6 

12 

2.820 

1.880 

" 

0.962 

1.01:4 

13 

2.598 

1.732 

" 

" 

0.890 

1.780 

14 

2.406 

1.604 

M 

0.826 

1.C.52 

15 

2.238 

1.492 

'• 

" 

0.772 

1.544 

16 

2.088 

1392 

" 

0.726 

1.452 

17 

1.962 

1.308 

" 

" 

0.684 

1.3(58 

18 

1.842 

1.228 

" 

•' 

0.648 

1.296 

19 

1.740 

1.160 

M 

" 

0.614 

1.228 

23 

1.650 

1.100 

" 

" 

0.584 

1.168 

21 

1.560 

1.040 

'• 

0.558 

1.116 

22 

1.482 

0.988 

" 

•' 

0.534 

1.068 

23 

1.410 

0.940 

1 

•' 

0.512 

1.024 

24 

1.350 

0.900 

41 

0.490 

0.980 

25 

1.284 

0.856 

" 

0.472 

0.944 

26 

1.230 

0.820 

0.456 

0.912 

27 

1.176 

0.784 

" 

0.440 

0.880 

28 

1.128 

0.752 

' 

" 

0.426 

0.852 

29 

1.080 

0.720 

' 

' 

0.412 

0.824 

30 

1.038 

0.692 

' 

' 

0.400 

0.800 

31 

0.996 

0.664 

" 

0.388 

0.776 

32 

0.960 

0.640 

« 

' 

0378 

0.756 

33 

0.924 

0.616 

" 

' 

0.368 

0.736 

34 

0.888 

0.592 

" 

• 

0.358 

0.716 

35 

0.858 

0.572 

" 

0.348 

0.696 

36 

0.828 

0.552 

" 

i 

0.340 

0.680 

37 

0.798 

0.532 

" 

0.332 

0.664 

38 

0.768 

0.512 

;' 

0.324 

0.648 

39 

0.738 

0.492 

" 

' 

0.318 

0.636 

40 

0.714 

0.476 

M 

' 

0.312 

O.C24 

41 

0.690 

0.460 

" 

' 

0.304 

0.608 

42 

0.666 

0.444 

« 

0.298 

0.596 

43 

0.642 

0.428 

" 

0.292 

0.584 

44 

0.618 

0.412 

" 

' 

0.288 

0.576 

45 

0.600 

0.400 

" 

0.284 

0.568 

STRAINS    IN    TRUSSE.S 


121 


V 

t7-  T 

Pi  -21 

Cj—  2*2 

% 

<74 

e5 

1 

T. 

T4 

T-, 

5 

8.5G8 

7.G16 

4.760 

0.1  66 

0.250 

0.416 

0.952 

2.856 

4.760 

G 

7 

7.123 
G.102 

6.336 
5.424 

3.960 
3.390 

n 

u 

0.680 

2.379 
2.041 

3.965 
3.402 

8 

5.337 

4.744 

2.965 

'; 

M 

" 

0.596 

1.788 

2.980 

g 

4.023 

4.200 

2.G25 

" 

" 

0.530 

1.590 

2.650 

10 

4.218 

3.776 

2.360 

" 

" 

M 

0.478 

1.434 

2.390 

11 

3.852 

3.424 

2.140 

" 

" 

0.435 

1.305 

2.175 

12 

3.519 

3.128 

1.955 

' 

" 

0.399 

1.197 

1.995 

13 

3.240 

2.880 

1.800 

" 

' 

•' 

0.369 

1.107 

1.845 

11 

3.006 

2.672 

1.670 

" 

" 

0.343 

1.029 

1.715 

15 

2.799 

2.488 

1.555 

'• 

0.320 

0.960 

1.600 

1C 

2.610 

2.320 

1.450 

<; 

" 

0.301 

0.903 

1.505 

17 

2.448 

2.176 

1.360 

" 

1 

" 

0.284 

0.852 

1.420 

18 

2.304 

2.048 

1.280 

'• 

" 

0.269 

0.807 

1.345 

if) 

2.1G9 

1.928 

1.205 

" 

'• 

" 

,    0.255 

0.765 

1.275 

ft) 

2.001 

1.832 

1.145 

'• 

" 

" 

0.242 

0.726 

1.210 

21 

1.944 

1.728 

1080 

" 

' 

" 

0.231 

0.693 

1.155 

22 

1.854 

1.G48 

1.030 

'' 

" 

0.221 

0.663 

1.105 

2:> 

1.7G4 

1.568 

0.980 

« 

" 

0.212 

0.636 

1.060 

21 

1.G83 

1.496 

0.935 

" 

' 

" 

0.203 

0.609 

1.015 

23 

1.G02 

'   1.424 

0.890 

" 

1 

" 

0.196 

0.588 

0.980 

2; 

1.539 

1.368 

0.855 

" 

" 

0.189 

0.567 

0.945 

27 

1.4G7 

1.304 

0.815 

" 

" 

« 

0.182 

0.546 

0.910 

28 

1.404 

1.248 

0.780 

'• 

t 

0.177 

0.531 

0.885 

29 

1.350 

1.200 

0.750 

" 

" 

• 

0.171 

0.513 

0.855 

30 

1.2)6 

1.152 

0.720 

" 

" 

0.166 

0.498 

C.830 

:>L 

1.242 

1.104 

0.690 

M 

•  ' 

' 

0.161 

0.483 

0.805 

32 

1.197 

1.0G4 

0.665 

" 

" 

' 

0.156 

0.468 

0.780 

33 

1.152 

1.024 

0.640 

M 

' 

0.152 

0.45G 

0.760 

34 

1.107 

0.984 

0.615 

" 

<; 

' 

0.148 

0.444 

0.740 

35 

1.071 

0.952 

0.595 

M 

0.144 

0.432 

0.720 

30 

1.035 

0.920 

0.575 

M 

" 

' 

0.141 

0.423 

0.705 

37 

0.999 

0.888 

0.555 

M 

« 

' 

0.138 

0.414 

0.690 

88 

0.954 

0.848 

0.530 

M 

« 

' 

0.134 

0.402 

0.670 

39 

0.918 

0.816 

0.510 

H 

0.132 

0.396 

0.660 

40 

0.891 

0.792 

0.495 

i- 

t 

0.129 

0.387 

0.645 

41 

0.8G4 

0.768 

0.480 

' 

0.126 

0.378 

O.C30 

42 

0.823 

0.736 

0.460 

M 

a 

0.123 

0.369 

0.615 

4:5 

0.801 

0.712 

0.445 

" 

0.121 

0.363 

0.605 

44 

0.774 

0.688 

0.4'JO 

" 

« 

M 

0.119 

0.357 

0.595 

45 

0.747 

0.664 

0.415 

" 

" 

" 

0.118 

0.354 

0.590 

122  STRAINS   IN   TRUSSED   BEAMS. 


STRAINS  IN  TRUSSED  BEAMS. 

When  a  beam  supported  at  the  ends,  is  required  to  carry  a 
greater  load  than  its  given  capacity,  and  trussing  is  resorted  to, 
it  may  become  necessary  to  find  what  portion  of  the  load  is  borne 
by  the  different  members  of  the  trussed  beam. 

Reference. 

Let  IF  =  Load  acting  on  truss  at  a  supported  point.  (See  figure.) 
Wl=  That  portion  of  IFacting  on  diagonals. 
W.2=  That  portion  of  IF" acting  on  beam. 
Ai=  Sectional  area  of  diagonal. 
A2=  Sectional  area  of  beam. 

El=  Modulus  of  elasticity  of  material  in  diagonals. 
E2=  Modulus  of  elasticity  of  material  in  beam. 

a  =  Length  of  diagonal. 

b  =  Distance  between  center  of  beam  and  point  of  support. 

c  =  Distance  between  abutment  and  point  of  support. 

{  =  Depth  of  beam. 
=  Depth  of  truss. 
/  =  Distance  between  center  of  beam  and  abutment. 

[NOTE. — Use  the  same  unit  of  length  and  weight.] 

No.  1. 

Fig.   209 


a3    '    /2     •    A2        E2 

—  . -/I.  A..  JIM 

Wj.       a3       /2<42        ^2 


STRAINS   IN   TEUSSED   BEAMS. 


-  .  TF 


When  load  is  equally  distributed  W  becomes  |  TF. 
No.  2. 


Fig.  210. 


211. 


W<2          2        a 


123 


2     '     a3    *    /2    '    A, 
>m       a3         /2         ^2 


A, 


2  Wi       a3        /2 


a3    *    /2 


TFl= 


"n7      "" 


PP, 


+  1 


When  load  is  equally  distributed  W  becomes  f  TF. 


124 


STRAINS   IN   TRUSSED   BEAMS. 


No,  3. 

Fig.  212. 


Al 


W2 


a  (a*  X 


_ 
h*    '   (V—b^c   '        El 


wl 


wl 


*—.  w 
+1 


When  load  is  equally  distributed  IF  becomes  f  PP". 


STRAINS   IN    TRUSSED    BEAMS. 


125 


No.  4. 

Figs.  213  and  214. 


h*       (I2  -  62)  c       Al        El 


2/2   •  a(a2+6c)  ' 


^        ±_ 
"         ''  ' 


A 

'    A, 


/2 
~~ 


h*        (P—b*)c 


'2WL  '    /2     '  a(a2+6c)  ' 


Wl 


•—  .  W 


W 


-+1 


When  load  is  equally  distributed  TF'becomes  f  W. 


126  STRAINS    IN    TRUSSES  WITH   PARALLEL   BOOMS. 

STRAINS  IN  TRUSSES,  WITH  PARALLEL  BOOMS. 
(Caused  by  Static  and  Moving  Loads) 

The  strain  in  the  upper  boom  is  always  compressive. 
*  The  strain  in  the  lower  boom  is  always  tensile. 

All  braces  inclined  down  from  the  nearest  abutment  are  in 
tension. 

All  braces  inclined  up  from  the  nearest  abutment  are  in  com 
pression. 

The  strains  in  the  verticals  and  diagonals  increase  from  the 
center  of  truss  to  abutment. 

The  strains  in  the  booms  decrease  from  the  center  of  truss  to 
abutment. 

A  moving  load,  advancing  over  a  truss,  &c.,  causes  the  maxi 
mum  moment  of  rupture  (which  under  an  equally  distributed 
load  is  at  the  center  of  truss)  to  shift  to  one  side  of  the  center, 
thereby  changing  the  nature  and  amount  of  strain  in  web  only. 
This  requires  either  the  enlargement  of  those  members  consti 
tuting  the  web  or  the  addition  of  so-called  counters,  (braces, 
struts,  or  ties.) 

To  find  the  point  from  center  of  truss  to  where  the  addition  ol 
counters  must  commence,  the  following  formula  is  used : 

Let  d  =  Distance  from  center  of  truss  to  point  where, 
maximum  moment  of  rupture  occurs,  and  where 
counter  bracing  must  commence. 

d/=  Distance  from  nearest  abutment  to  ditto. 


Anad/=J d  =  -^- 

2  W/ 

These  results  will  be  found  to  agree  with  formulas  for  "  Counter 
Strains"  when  Vm  becomes  negative. 

Reference. 

N  =  Total  number  of  bays  in  a  truss. 
HK  =  Horizontal  strains  in  booms. 

Fn  =  Strains  in  verticals. 

yn  =  Strains  in  diagonals. 

Vm  =  Vertical  strains  acting  on  counters  Ym. 

Ym  =  Strains  in  counters,  opposite  in  kind  to  Yn. 


STRAINS   IN   TRUSSES  WITH    PARALLEL   BOOMS.  127 

IF  =  Weight  of  static  load,  equally   distributed    over   whole 

length  of  truss. 

W,=  "Weight  of  moving  load,  equally  distributed  over  whole 
length  of  truss. 

h  =  Height  or  depth  of  truss  between  the  center  of  gravity  of 
booms. 

I  =  Span  or  length  of  truss  from  abutment  to  abutment. 

n  —  Number  of  member,  counting  from  abutment  A. 
•in  =  Number  of  member,  between  center  and  abutment  B. 

r  =  Half  the  length  of  a  panel  or  bay. 

s  =  Length  of  a  panel  or  bay. 

w  =  Weight  of  static  load  per  unit  of  length  I. 
iv/=  Weight  of  moving  load  per  unit  of  length  I. 

v  =  Angle  between  horizontal  and  diagonal. 
For  other  designations,  see  diagrams  and  examples. 

The  angle  v  for  Howe  Truss  is  generally  45°. 
The  angle  v  for  Whipple  Truss  is  generally  45°. 
The  angle  v  for  Lattice  Truss  is  generally  45°. 
The  angle  v  for  Warren  Truss  is  generally  60°. 

The  proportion  of  height  h  to  span  I  is  from  4  to  -^  gener 
ally  Tv 


128 


STRAINS    IN   TRUSSES  WITH    PARALLEL   BOOMS. 


STRAINS   IN"   TRUSSES  WITH    PARALLEL    BOOMS.  129 

HOWE  TRUSS.     (Figs.  215,  216,  217,  and  218.) 

Additional  Reference. 

,Tn—  Distance  from  abutment  A  to  center  of  bay. 
yn—  Distance  from  abutment  A  to  apex  of  bay. 

Static  or  Permanent  Load,  equally  distributed  over  whole  length  of 
Truss, 

Strains  in  Booms. 

w  w 


Strains  in  Verticals. 

w      w 
F.=  -  ---  -*. 

Strains  in  Diagonals. 
^n=  Vn  cosec.  v. 


Moving  and  Static  Load,  each  equally  distributed  per  unit  of 
length. 


•~ 


Strains  in  Booms. 

w+w, 


~2h~'  •    °  2hl~  • 

Strains  in  Verticals. 


Strains  in  Diagonals. 
Yn  =  F"n  cosec.  v. 

Strains  in  Counters. 


130  STRAINS   IN    TRUSSES    WITH    PARALLEL   BOOMS. 

EXAMPLE.     (Figs.  215,  216,  217,  and  218.) 

Moving  Load,  (as  railway  train  passing  over  bridge.) 
We  will  assume  W  =  50,000  Ibs. 
Wt=  100,000  Ibs. 
I  =  100  feet. 
h  =  10  feet. 
v  =  45°,  (cosec.  =  1.414.) 

Horizontal  Strains  in  Booms,  (compression  inupper,  tcnsionin  lower.) 

W-\-  Wi  JF  +  TFi        2  __  50UOO+  100000 

//a  =          2/i         ' y"  ~          2hl        ' 2/n  =  ~20 

50000  +  100000_      2  __  _  ^5      2 

yn~~  2000         ~~'y&  ~~  '       'J/a         0-2/n 

//!  =  7500 . 10  —  75 . 100    =  67,500  Ibs. 
II 2  =  7000.20  —  75.400    =  120,000  Ibs. 
11. 3  =  7000.30  —  75.900    =  157,500  Ibs. 
7/t  =  7500.40  —  75.1600  =  180,000  Ibs. 
J15  =  7500.50  —  75.2500  =  187,500  Ibs. 

Strains  in  Verticals. 

W         W  Wl  ,\2__  50022__  5000° 

n          i>  r '**  "*"  'W  ^    ".  ^  2  '   100 

.(^  —  .rj 2  =  25000  —  500. .TnH 


Strains  in  Figs.  215   210   217   218 

F1  =  25000  — 500.5    +5. 05*= 67625  Ten.  Ten.  Corn.  Com. 
F2  =  25000  — 500. 15  4-5. 852  =  53625     " 
F,  =  25000  — 500. 25+5. 7r>a  =  40625      " 
F4  =  i)5000— 500. 35  +  5.  ()52=  28625      " 
Vl  =  25000  —  500. 45 +o.552  =  17625      " 

Counter  Strains  (Vm)  for  Strains  in  Counters. 
V6  =  20000  —  500.55  +  5.45*  =  7625. 
F7  =  25000  —  500.65  +  5.85'  =  5625. 

Strains  in  Diagonals. 
Yn  =  Fn  cosec  v. 

Strains  in  Figs.    215         210         217         218 

y\  =  67625  .  1.414  =  1)5,620  Ibs.    Com.    Com.    Ten.    Ten. 
Y2  ==  53625  .  1.414  =  75,826  Ibs. 
F3  =  40625  .1  414  =  57,44  i  Ibs. 
F4  =  28625  .  1.4 14  =  40,476  Ibs.       " 
F5  =  17625  .  1.414  =  24,922  Ibs. 

Strains  in  Counters,  (dotted  lines,  Fig.  215,  for  example.) 
Yn  =  Fm  cosec.  v. 

Strains  in  Figs.    215         210         2L7          218 

FG  =  7625  .  1.414  ^r  10,762  Ibs.     Com.    Com.    Tun.      Ten. 
F7  =5625  .  1.414=    7,954  Ibs. 


STRAINS   IN   TRUSSES  WITH   PARALLEL   BOOMS. 


131 


Pig.  219.  LATTICE  TRUSS  WITH  VERTICAL  NUMBERS. 

'sssvs. 

Fig.  219.     Load  on  either  Boom. 

To  compute  the  strains  in  this  truss,  the 
easiest  method  is  to  find  the  values  of  /fn,  Fn, 
Fm,  Fn,  and  Fm  for  a  Howe  Truss,  (Figs.  215, 
(  216,  217,  and  218  )  loaded  in  the  same  man- 
|  ner,  (upper  or  lower  boom.)    These  values  in 
the  following  formulas  for  the  above  truss  will 
i  give  the  required  strains: 


Strains  in  Booms.     (8.) 


<?!=- 


4*2~T"  jC*8  /^  n        r/  -"n  l~ 

: ,  generally  on=:  —     — f— 

Strains  in  Verticals.     (U.) 
Upper  boom  loaded — compression. 
Lower  boom  loaded — tension. 


Wl 


constant. 


Strains  in  End  Post     (  U0.) 

Upper  boom  loaded. 
U0=  U  -f-  ^1=  compression. 

Lower  boom  loaded. 
U0=  Si^=  compression. 

Strains  in  Diagonals.     (D.) 


D*=^~ 


I?_ 

2 
Y* 


Generally  Dn= 

i 

Strains  in  Counters. 
Generally  Z>m=  — 


132  STRAINS   IN  TRUSSES  WITH   PARALLEL  BOOMS. 

Fig.  220.  WARREN  TRUSS. 

Fig.  220.     Lower  Boom  Loaded. 

Additional  Reference. 
#n  =  Distance  from  abutment  A  to  center 

of  diagonal. 
3/n  =  Distance  from  abutment  A  to  apex 

of  bay  of  upper  boom. 
2n  =  Distance  from  abutment  A  to  apex 

of  bay  of  lower  boom. 

Static  or  Permanent  Load,  equally  distributed 

over  whole  length  of  Truss. 

Strains  in  Booms. 

Upper. 

77  W  W       2 

^=-ir°-2Ar*°2 

Lower. 

„        W  W 


Strains  in  Verticals. 

K  =  —  ---  —  a?n     (  Fn  acts  at  the  end 

-  I 


of  arn.) 


Strains  in  Diagonals. 
Y^  ==.  Fn  cosec.  v. 


Moving  and  Static  Load,  each  equally  dis 
tributed  per  unit  of  length. 

Strains  in  Booms. 
Upper. 


Lower. 


Strains  in  Verticals. 

«--?-?•  +5  " 

Strains  in  Diagonals. 
Y*  =  Fn  cosec  v. 


STRAINS  IX  TRUSSES   WITH   PARALLEL   BOOMS.  133 

Strains  in  Counters. 

EXAMPLE.     (Fig.  220.) 
Moving  Load  (as  railway  train  passing  over  bridge)  on  lower  Bocm. 

We  will  assume  W  =  50,000  Ibs. 
Wi=  100,000  Ibs. 
I  =  100  feet. 
A  =  10  feet. 
?;  =  630  20',  (cosec.  =  1.12.) 


_ 


Horizontal  Strains  in  Upper  Boom.     (Compression.) 
W+Wi    ^  2  __    50000  +  100000 


2h  2hl  2.10 

50000  +  100000    2  _  150000 
Zn~~    2.10  .  100    '**  ~   ~~20   '2n~~ 


HI  =  7500 . 10  —  75 . 100  =  67,500  Ibs. 
#2  =  7500.20  —  75.400  =  120,000  Ibs. 
H3  =  7500.30  —  75.900  =  157,500  Ibs. 
H4  =  7500.40  —  75. 1600  =  180,000  Ibs. 
H6  =  7500.50  —  75.2500  =  187,500  Ibs. 

Horizontal  Strains  in  Lower  Boom.     (Tension.) 

i  JF-f  Jh  50000  +  100000 

" 


2A         2hl  2.10 

50000  -f-  100000  f  2_  150000       150000 
2.10  .~100~   -2/n  ~    20   '^n     2000~ 


—75.25  =  37500—  1875=  35,625  Ibs. 
#2  =  7500.15  —  75.225  =112500—  16875=  95,6251bs. 
H3  =  7500 . 25  —  75 . 625  =  1 87500  —  46875  =  140,625  Ibs. 
Jff4  =  7500.35  —  75. 1225  =  262500  —  91875  =  170,625  Ibs. 
H6  =  7500.45  —  75.2025  =  337500  —  151875  =  185,623  Ibs. 


134 


STRAINS   IN   TRUSSES    WITH    PARALLEL   BOOMS. 


Strains  in  Verticals. 
YU  =  Fn  cosec.  v. 

W         W  W,    ,  50000 

Fn  =  — —  .rrn  +  -nT-(J  —  *»)  =  — o — 

2  £  2£  2 


50000 

"Too" 


rn  +  ^^T-(100  —  *u)2  =  25000  —  500,rn  +  5.  (100  —  xu)* 


v, 

ss 

25000  — 

500 

.  2.5  + 

5  . 

,  9506.25 

= 

71281.25. 

r. 

5= 

25000  — 

500 

.  7.5  + 

5  , 

,  8556.25 

= 

64031.25. 

F~ 

S=9 

25000  — 

500 

.  12.5  + 

5  . 

7656.25 

= 

57031.25. 

P 

as 

25000  — 

500 

.  17.5  + 

5  , 

,  6806.25 

= 

50281.25. 

F8 

— 

25000  — 

500 

.  22.5  + 

5  , 

.  6006.25 

= 

43781.25. 

^« 

= 

25000  — 

500 

.  27.5  + 

5 

,  5256.25 

— 

37531.25. 

[J_ 

— 

25000  — 

500 

.  32.5  + 

5  . 

.  4556.25 

= 

31531.25. 

P- 

_-r^= 

25000  — 

500 

.  37.5  + 

5 

.  3906.25 

— 

25781.25. 

p* 

— 

25000  — 

500 

.  42.5  + 

5  , 

.  3306.25 

SBB 

20281.25. 

l'n 

(== 

25000  — 

500 

.  47.5  + 

5  , 

.  2756.25 

= 

14031.25. 

F!  j=  25000 - 
Fj  2=  25000  - 
F!  3=  25000- 


=  71281.25  , 
=  64031.25 
=  57031.25 
=  50281.25  , 
=  43781.25 
=  37531.25  , 
=  31531.25 
=  25781.25 
=  20281.25 
n=  14031.25  , 


Counter  Strains.     ( Fm.) 

•  500  .  52.5  4-  5  .  2256.25  = 

•  500  .  57.5  +  5  .  1806.25  = 

•  500  .  62.5  +  5  .  1406.25  = 

Strains  in  Diagonals. 
Y"n=  F"n  cosec.  v. 


10031.25. 

'•    528125. 
781.25. 


1.12  =  79,835  Ibs. 
1. 12  =  71, 715  Ibs. 
1.12  =  63,875  Ibs. 
1.12  =  56,315  Ibs. 
1.12  =  49,035  Ibs. 
1.12  =  42,035  Ibs. 
1.12  =  35,315  Ibs. 
1.12  =  28,875  Ibs. 
1.12  =  22.715  Ibs. 
1.12  =  15,715  Ibs. 


Compression  in  Yl  and  F2, 
Tension  in  Y2  and  Y19. 
Compression  in  Y3  arid  Yli 
Tension  in  Y±  and  ]T17. 
Compression  in  Y5  and  Y1 
Tension  in  Y6  and  Y15. 
Compression  in  F"7  and  yr 
Tension  in  Ys  and  ]T13. 
Compression  in  Yg  and  Yl 
Tension  in  Ylo  and  Yll. 


Counter  Strains. 


=  T/m  COS6C- 


FH=  10031.25  .  1.12  =  11,235  Ibs.  Compression  in 
F12=  5281.25.1.12=  5,915  Ibs.  Tension  in  Y9  i 
F13=  781.25  .  1.12=  875  Ibs.  Compression  in 


i  and  '. 


lif'j 


STRAINS    IN   TRUSSES  WITH    PARALLEL    BOOMS.  135 

Fiy.  221.  WARREN  TRUSS. 

Fig.  221.      Upper  Boom  Loaded. 

Additional  Reference. 
=  Distance  from  abutment  A  to  center 

of  bay  of  upper  boom. 
2/n  =  Distance  from  abutment  A  to  apex  of 

bay  of  upper  boom. 

2n  =  Distance  from  abutment  A  to  apex  of 
bay  of  lower  boom. 

Static  or  Permanent  Load,  equally  distributed 
over  whole  length  of  Truss. 


Strains  in  Booms. 
Upper. 
/  W 


Wr* 


Lower. 


w 


2h    •*"" 


Strains  in  Verticals. 
_     W  W 

Ta  —   —  J—    •    #n 

£i  I/ 

Strains  in  Diagonals. 
Yn  =  Va  cosec.  v. 

p  Moving  and  Static  Load,  each  equally  dis 
tributed  per  unit  of  length. 

Strains  in  Booms. 
Upper. 

w+Wl          (w  +  w, 

^       2n,        '    a"  (      2hl       '   *  H 


2hl 
Lower. 


TT+TF, 


136  STRAINS   IN   TRUSSES   WITH    PARALLEL   BOOMS. 

Strains  in  Verticals. 


- 

Strains  in  Diagonals. 
Tn  =  Fn  cosec.  v. 

Strains  in  Counters. 

w       w         w 

—  ---     *-^""(l~*' 


=  7m  cosec. 


EXAMPLE.     (Fig.  221.) 
Moving  Load  (as  railway  train  passing  over  bridge)  on  Upper  Boom. 

We  will  assume  W  =  50,000  Ibs. 
Wl=  100,000  Ibs. 
I  =  100  feet. 
h  =  10  feet. 
v  =  63°  20',  r  =  5  feet. 

Horizontal  Strains  in  Upper  Boom.     (Compression.) 
jT+JFi  rjF+TPi      2,(W+W})r*-}_ 

2/i          n       L      2M      <2a  H  2AZ         J  "" 

lupOOO_  ^  2         150000.  5*  1  _ 
~200(r~'Zn  ^          ^000       J  ~ 


J50000_  rluOOO 

20       'Za 

7500.  2n—  [75.2n2+  1875] 


jff1==7500.5    — 
H2=  7500.15  — 
,=  7500.25  — 


75.25      -f  1875" 
75.225    +  1875: 
75.625    -f  1875 
' 


H±=  7500.35  —  [75.1225  -j-  1875 
H5=  7500.45  —  [75.2025  +  1875; 


=  33,750  Ibs. 
=  93,750  Ibs. 
=  138,750  Ibs. 
=  168,750  Ibs. 
=  183,750  Ibs. 


Horizontal  Strains  in  Lower  Boom     (Tension.) 


-  . 

^==7500.10  —  75.100  =  67,500  Ibs. 
H,=  7500.20  —  75.400  =  120,000  Ibs. 
Hl=  7500.30  —  75.900  =  157,500  Ibs. 
J/4==  7500.40  —  75.1600  =  180,000  Ibs. 
JI5=  7500.50  —  75.2500  =  187,500  Ibs. 


STRAINS  ITS  TRUSSES  WITH   PARALLEL  BOOMS.  137 

Strains  in  Verticals. 

Fn  =  -J---^.*  +  _|L(Z_a;n)2  =  25000-500.*  + 
5.(Z  — zn)2 

F1==  25000  —  500.5  +  5.952  =  67,625  Ibs. 
F2=  25000  —  500.15  +  5.85*  =  53,625  Ibs. 
F3==  25000  —  500.25  +  5.752  =  40,625  Ibs. 
F4=  25000  —  500.35  +  5.652  =  28,625  Ibs. 
F5=  25000  —  500.45  +  5.552  ==  17,625  Ibs. 

Counter  Strains. 
FG=  25000  —  500.55  +  5.45*  =  7,625  Ibs. 

Strains  in  Diagonals. 

Fn  =  Fn  cosec. 
Y1  =  67625  .  1.12  =  75,740  Ibs.      Tension  in  Yl  and  F10; 

compression  in  Fa  and  YA. 
Y2  =  53625  .  1.12  =  60,060  Ibs.      Tension  in  Y2  and  F9  ; 

compression  in  Fb  and  Fb. 
F3  =  40625  .  1.12  =  45,500  Ibs.      Tension  in  F3  and  Y6- 

compression  in  Yc  and  Y0. 
F4  =  28625  .  1.12  =  32,060  Ibs.      Tension  in  F4  and  77 ; 

compression  in  Fd  and  Fd. 
F5  =  17625  .  1.12  =  19,740  Ibs.      Tension  in  F5  and  F6 ; 

compression  in  Fe  and  Fe. 

Counter  Strains. 
Fm  =  Fm  cosec.  v. 

F6  =  7625  .  1.12  =  8,540  Ibs.     Compression  in  F5  and  F6 ; 
tension  in  Ffl  and  F. 


133 


STRAINS    IN    TUUSSES   \VIT11    PARALLEL    BJOMS. 


8TEAINS   IN   TRUSSES  WITH   PARALLEL   BOOMS.  139 

LATTICE  TRUSS.     (Figs.  222,  223,  and  224.) 
Lower  Boom  Loaded. 

Additional  Reference. 
r=  Half  the  length  of  a  bay   of  simple   truss.     (Figs.   222 

and  223.) 

xn=  Distance  from  abutment  A  to  center  of  bay  of  lower  boom. 
2/n—  Distance  from  abutment  A  to  apex  of  bay  of  upper  boom. 
zn=  Distance  from  abutment  A  to  apex  of  bay  of  lower  boom, 

The  formulas  are  for  the  strains  in  the  simple  trusses,  (Figs. 
222  and  223.)  Fig.  224  shows  the  simple  trusses  combined,  con 
stituting  the  Lattice  Truss.  . 

When  the  upper  boom  is  loaded,  treat  the  strains  as  acting  up 
ward  and  the  truss  inverted:  the  strains  will  be  of  the  same 
amount  in  each  member,  but  different  in  kind. 

Static  or  Permanent  Load,  equally  distributed  over  whole  length  of 
Truss. 


Strains  in  Booms. 
Upper. 

H—    W      (     4-     T   \  -     W       (-I-    T   V-U  Wr2 

u~  "2A  '  rn+  TV  ~~  2/iI  '  VB~*"  ~2~)  +  W 

Lower. 

_  W      f  r  \         W      f  r  \2        STFr2 

'n~~^"  '  V^u         2  /        2/iI  '  \^m       2  /  ~       8/ti 
Strains  in  Verticals. 

Fn  =  "4  2T  '  *n 

Strains  in  Diagonals. 


Moving  and  Static  Load,  each  equally  distributed  per  unit  of 

length. 
Strains  in  Booms. 

Upper. 
n_W+W±(.i      r   \        W+Wi    (     ,      r    \2      (TF+TFi)r2 

Lower. 

//_TF-f-IFL/    __L\___^±I^i  /    _  r   V-     3(ir-h)rTF* 


140  STRAINS   IN   TRUSSES   WITH    PARALLEL   BOOMS, 

Strains  in  Verticals. 
W         W  Wi 


Strains  in  Diagonals. 
Yn  =  Vr,  cosec  v. 

Strains  in  Counters. 
W         W  Wl 


=  Fm  cosec<  v- 


[NOTE.— The  strains  in  Fa.b.c,  ....  are  equal  in  amount,  but  different 
in  kind  to  the  strains  in  H,2, 3,  .... 

EXAMPLE.     (Figs.  222,  223,  and  224.) 

Moving  Load  (as  railway  train  passing  over  bridge)  on  Lower  Boom. 

We  will  assume  W  =  50,000  Ibs. 
Wi=  100,000  Ibs. 
I  =  100  feet. 
h  =  10  feet. 
v  =  63°  20',  (cosec.  =  1. 12.)  r  =  5  feet.. 

Horizontal  Straws  in  Upper  Boom.     (Compression.     Fig.  224.) 


•  =  7500(zn+  2.5)— -75(2n+  2.5)2  +  468.75 

HQ=  7500  .  (  0  +  2.5)  —  75  .  (  0  +  2.5)2  +  468.75  =  18,750  Ibs. 
£l=  7500  .  (  5  +  2.5)  —  75  .  (  5  +  2.5)2  +  468.75  =  52,500  Ibs. 
//„=  7500  .  (10+  2.6)  —  75  .  (10  +  2.5)2  +  468.75  =  82,500  Ibs. 
/£=  7500  .  (15  +  2.5)  —  75  (15  +  2.5)2  +  468.75  =  108,750  Ibs. 
JI4=  7500  .  (20  +  2.5)  —  75  .  (20  +  2.5)2  +  468.75  =  131,250  Ibs. 
//.=  7500  .  (25  +  2.5)  —  75  .  (25  +  2.5)2  +  468.75  =  150,000  Ibs. 
l/(.=  7500  .  (30  +  2.5)  —  75  .  (30  +  2.5)2  +  468.75  =  165,000  Ibs. 
H7=  7500  .  (35  +  2.5)  —  75  .  (35  +  2.5)2  +  468.75  =  176,250  Ibs. 
/78==7500  .  (40+2.5)  — 75  .  (40+  2.5)2+ 468.75^  183,750  Ibs. 
Hf=  7500  .  (45  +  2.5)  —  75  .  (45  +  2.5)2  +  458.75  ==  187,500  Ibs. 


STRAINS   IN   TRUSSES  WITH   PARALLEL   BOOMS. 


141 


Horizontal  Strains  in  Lower  Boom.     (Tension.     Fig.  224.) 


_ 


-na=  - 

2h   '  Vn   I 

1  1     2hl   '  Vyu" 

3(17+  i 

7\t« 

V               75QO    (y 

,—  2.5)  —  75  .  (yn—  2.1 

Shi 

fli  =  7500 

.(  5_25)  —  75. 

(  5  —  2.5)2—1406.25  = 

H2  =  7500  , 

,(10  —  2.5)  —  75. 

(10  —  2.5)2—1406.25  = 

#,  =  7500 
H\  =  7500 

.  (15  —  2.5)—  75. 
.(20  —  2.5)—  75. 

(15  —  2.5)2  —  1406.25  = 
(20—2.5)2—1406.25  = 

H5  =  7500  , 

.(25  —  2.5)—  75. 

(25—2.5)2—1406.25  = 

H6  =  7500  , 

,  (30  —  2.5)—  75. 

(30—2.5)2—1406.25  = 

HI  =  7500 

.(35  —  2.5)—  75. 

(35—2.5)2—1406.25  = 

HS  =  7500 

.(40  —  2.5)  —  75. 

(40—2.5)2—1406.25  = 

£T9  =  7500 
#10=  7500 

.(45  —  2.5)—  75. 
.(50  —  2.5)—  75. 

(45_  2.5)2—  1406.25  = 
(50—2.5)2—1406.25  = 

2.5) 2  —  1406.25 


16,875  Ibs. 

50,625  Ibs. 

80,625  Ibs. 
106,875  Ibs. 
129,375  Ibs. 
148.125  Ibs. 
163,1  25  Ibs. 
174,375  Ibs. 
181,875  Ibs. 
185,625  Ibs. 


SIMPLE  TRUSS.     (Fig.  222.) 
Strains  in  Verticals.     (Fn.) 


"T 


IT 


-  (l  ~  **)2=  125°°  - 

2.5  .(Z-*n)2 


II  II  II  II  II 

tTuV^V 

12500 
12500 
12500 
12500 
12500 

—  250  , 
—  250 
—  250  , 
—  250  . 
—  250  . 

0  + 

.10  + 
,  20  + 
30  + 
40  + 

2.5  . 
2.5  . 
2.5  . 

2.5  . 

2.8  . 

1002 
902 

802 
702 
602 

= 

37,250  Ibs. 
30,250  Ibs. 
22,500  Ibs. 
17,250  Ibs. 
11,  500  Ibs. 

Com.  in  U. 


Counter  Strains.     (Fm.) 

F6=  12500  —  250  .  50  +  2.5  .  502  =  6,250  Ibs. 
F7=  12500  —  250  .  60  +  2.5  .  402  =  1,500  Ibs. 


Y1 


Strains  in  Diagonals. 

Yn=  Vn  C0sec- 

Tension  in  Y1  and  F10; 


37250  .  1.12  =  41,720  Ibs. 
compression  in  Fa  and  Y» 


Y2  =  30250  .  1.12  =  33,880  Ibs. 
compression  in  Y*  and  Y^. 


Tension  in  Y2  and  F9 


142  STRAINS   IN   TRUSSES   WITH    PARALLEL   BOOMS. 


compression  in  Ye  and  Y0. 
F4  =  17250  .  1.12  =  19,320  Ibs.  Tension  in  F4  and  F7 ; 

compression  in  Fd  and  Fd. 
F5  =  11500  .  1.12  =  12,880  Ibs.  Tension  in  Y5  and  F6  • 

compression  in  Fe  and  Fe. 

Counter  Strains. 
Ym=  Vm  cosec.  v. 

F6  =  6250  .  1.12=  7,000  Ibs.     Compression  in  F5  and  F6 ; 
tension  in  Fe  and  Ye. 

=  1,680  Ibs.     Compression  in  F4  and  F7; 


SIMPLE  TRUSS.     (Fig.  223.) 
Strains  in  Verticals.     (  Fn.) 

l\—  12500  —  250  .  5  +  2.5  .  952  =  33812 .5. 
Vf=  12500  —  250  .  15  +  2.5  .  852  =  26812.5. 
T7  —  12500  —  250  .  25  +  2,5  .  752  =  20312.5. 
F4=  12500  —  250  .  35  +  2.5  .  652  ==  14312.5. 
F5=  12500  —  250  .  45  +  2.5  .  552  =  8812.5. 

Counter  Strains.     (Vm.) 
T7G=  12500  —  250  .  55  +  2.5  .  452  =  3812. 
Strains  in  Diagonals. 
Fn=  Vn  cosec.  v. 

F1=  33812.5  .  1.12  =  37,870  Ibs.    Compression  in  Y}  and  FIO: 

tension  in  Fa  and  Fa. 
Y,=  26812.5  .  1.12  =  30;030  Ibs.  Compression  in  F,  and  F{); 

tension  in  Fb  and  Fb. 
Y,—  20312  5  .  1.12  =  22,750  Ibs.  Compression  in  F,  and  Fs; 

tension  in  Fc  and  Fc. 
F4r=  14312.5  .  1.12  =  16,030  Ibs.  Compression  in  F4  and  F7: 

tension  in  Fd  and  Fd. 
F5=  8812  5  .  1.12  =  9,870  Ibs.  Compression  in  F5  and  Ffi; 

tension  in  Fe  and  Fe. 

Counter  Strains. 

Ym  =  Vm  cosec-  v- 

Y^-=  3812.5  .  1.12  =  4,270  Ibs.     Tension  in  F5  and  F6;  com 
pression  in  Fe  and  Fe. 


STRAINS    IN    TRUSSES  WITH    PARALLEL    BOOMS. 


143 


Fly.  225. 
Lower  boom  loaded. 


t 


144  STRAINS  IN  TRUSSES  WITH    PARALLEL   BOOMS. 

WHIPPLE  TRUSS.     (Figs.  225,  226,  227,  and  228.) 

Additional  Reference. 
#n,  yn=  Distance  from  abutment  A  to  end  of  bay. 


Static  or  Permanent  Load,  equally  distributed  over  whole  length  oj 
Truss. 

Strains  in  Booms. 
W  W  sW  sW 

a  =  "2JT  '  yB~  W  y»  +  ~2hT  '  **-  ~4T 

Strains  in  Verticals. 

T7-      W  W 

Kn~~l        2T 

Strains  in  Diagonals. 
Yn=  Vn  cosec.  v. 


Moving  and  Static  Load,  each  equally  distributed  per  unit  of 
length. 

Strains  in  Booms. 


2hl~'  '    *  2hl 

s(W+W1) 


Strains  in  Verticals. 
TF          TF 


Strains  in  Diagonals. 
Fn  =  Fn  cosec.  v. 

Strains  in  Counters. 


STRAINS   IN   TRUSSES   WITH   PARALLEL   BOOMS. 


145 


EXAMPLE.     (Figs.  225,  226,  227,  and  228.) 

(With  20  Bays.) 
Moving  Load,  (as  railway  train  passing  over  bridge.) 

Let  W  =  50,000  Ibs. 
Wl=  100,000  Ibs. 
I  =  100  feet. 
h  =  10  feet,  s  =  5  feet. 
v  —  45°.     (End  diagonals  v  =  26°  30'.) 

Horizontal  Strains  in  Booms.     (Compression  in  upper,  tension  in 
lower.) 


27. 

S(W  + 

"l)              h-crnn    ..             nr          o 

OfTK      a,          1 

i  ft 

T 

11  = 

7500  . 

0 

—  75.    O2  — 

375 

•yn  ~ 
•   o  + 

'    y\i  \ 

18750  = 

-  JLo 
18 

7C 

,7. 

RY= 

7500  . 

5 

—  75, 

.    52  — 

375 

.     5  + 

18750  = 

52 

,r>' 

JL= 

7500  . 

10 

—  75, 

.  102  — 

375  , 

,  10  + 

18750  = 

82 

,51 

|3= 

7500  . 

15 

—  75 

.  152  — 

375 

.  15  + 

18750  = 

108 

,7. 

7500  . 

20 

—  75  . 

202  — 

375  . 

20  + 

18750  = 

131 

,& 

//5= 

7500  . 

35 

—  75  . 

252  — 

375  , 

,  25  + 

18750  = 

150 

,CM 

H{~ 

7500  . 

30 

—  75  , 

.  302  — 

375  , 

,  30  + 

18750  = 

165 

,!)( 

H1= 

7500  . 

35 

—  75 

.  352  — 

375  , 

,  35  + 

18750  = 

176 

•)•, 

HX— 

7500  . 

40 

—  75  , 

,  402  — 

375  , 

,  40  + 

18750  = 

183 

i  '  < 

J/9= 

7500  . 

45 

—  75 

.  452  — 

375  , 

.  45  + 

18750  = 

187 

ft 

_  / 

jfefe^A-ii 

8* 

r" 

Strains  in  Verticals. 


~ 


•  =  75,000  Ibs. 


Fi= 

F2= 
F3= 
V±= 
F5= 
F6= 
F7= 


Strains  in  Figs.  2: 
12500  —  250  .    0  +  2.5  .  1002  =  37,500  Ibs.    C 
12500  —  250.    5+25.    952  =  338121bs      ' 
12500—250.10+2.5.    902  =  30,250  Ibs.     ' 
12500  —  250.15+2.5.    852=r  26,812  Ibs.     ' 
12500  —  250  .  20  +  2.5  .    802  =  23,500  Ibs.     ' 
19^nn       o^n     of\  i    o  o       ^7'~9       OA  01  o  i  u 

5  223  227   22 
.     C.     T.     T 

zouu  —  zou  .  zo  +  A.A  .    7o2  =  20,312  IDS. 

1  9':>r>n             9^0          QH      1       9  Pi             *7A2             1  n   Of^A  11                ( 

izouu  —  z,o(j  .  6(J  +  Z.o  .    702  =  17,250  Ibs. 
10 

146 


STRAINS  IN  TRUSSES  WITH   PARALLEL   BOOMS. 


Strains  in  Figs.  225  225  227  228 
Vs  =  12500  —  250  .  S5  +  2.5  .  652=  14,312  Ibs.    C.    C.    T.    T. 

Vg  =  12500  —  250  .  40+  2.5  .  602=  11,500  Ibs.     "     ' 

F10=  12500  —  250  .  45+  2.5  .  552  =    8, 812  Ibs.     "     "     <l      " 


Fn=  12500  — 250 
Fr,=  12500  — 250  , 
Fw=  12500  — 250  . 


Vm  Acting  on  Counters. 

,  50+2.5  .  502  =  6,250  Ibs. 
55+2.5  .  452  =  3, 812  Ibs. 
60+25  .  402=  1,500  Ibs. 

Strains  in  Diagonals. 


•Jn  —  Kn  cosec.  v. 

Strains  in  Figs.   225 

Yl  =  37500  .  1.117  =  41,887  Ibs.  Ten. 
F2  =  33812  .  1.414  =  47,810  Ibs. 
F8  =  30250  .  1.414  =  42,773  Ibs. 
F4  =  26812  .  1.414  =  37,913  Ibs. 
F5  =  23500  .  1.414  =  33,229  Ibs. 
F6  =  20312  .  1.414  =  28,722  Ibs. 
7,  =  17250  .  1.414  =  24,391  Ibs. 
F8  =  14312  .  1.414  =  20,238  Ibs. 
F9  =  11500  .  1.414  =  16,261  Ibs. 
*10=  8812  .  1.414  =  12,461  Ibs. 

Strains  in  Counters. 

Fu=  6250  .  1.414  =  8,837  Ibs. 
F19=  3812  .  1.414  =  5,391  Ibs. 
F,;=  1500  .  1.414  =  2,121  Ibs. 


223 

Ten. 


227 
Com. 


223 
Com. 


[NOTE. — If  counter  braces  are  not  inserted,  Vn,  F"i2,and  Tri3,andlr8> 
Yg,  and  Y\Q  will  have  an  additional  strain,  opposite  in  kind  and  equal 
to  V\  i,  V\  2,  and  V\  3,  and  Y\\,  ^12?  and  Y\  3  ;  but  if  counters  are  used, 
the  strain  V\\4  F"i2»and  Vis  will  n°t  occur  in  the  structure,  but  will 
be  necessary  to  determine  the  strain  in  FH,  Fi2,and  F\  3  only.  FH, 
FI  2 ,  and  FI  3  will  then  be  inclined  in  the  same  direction  as  the  diago 
nals  from  abutment  A  to  center  of  truss,  the  character  of  strain  being  the 
same.  (See  also  "Howe  Truss") 

Keep  in  mind  that  each  half  truss,  as  to  the  character  and  amount  of 
strain  in  the  respective  members,  is  alike.] 


STRAINS  IN  PARABOLIC  CURVED  TRUSSES.  147 


STRAINS  IN  PARABOLIC  CURVED   TRUSSES— "  BOW 
STRING  GIRDERS." 

(Figs.  229,  230,  231,  232,  233,  and  234.) 

The  strains  in  the  lower  boom  (when  horizontal)  are  the  greatest, 
and  equal  in  every  bay,  when  the  load  is  equally  distributed  over 
the  whole  length. 

The  strains  in  the  arch  or  upper  boom  are  also  greatest  when 
the  load  is  equally  distributed  over  the  whole  length;  the  strains 
gradually  increasing  from  the  middle  to  the  supports. 

The  strains  in  the  diagonals,  whether  single  or  double,  in  a 
bay  are,  when  the  load  is  equally  distributed,  everywhere  null. 
When  the  load  is  unequally  distributed,  and  one  diagonal  to  each 
bay  is  used,  they  will  be  either  in  compression  or  tension.  The 
character  of  the  maximum  of  strains  will  be  as  follows:  Assume 
the  left  half  of  truss  to  be  loaded.  All  diagonals  inclined  up  from 
left  to  right  abutment  are  in  tension;  if  inclined  down,  in  com 
pression.  The  character  of  strains  will  be  vice  versa  when  the 
right  half  only  is  loaded. 

The  strains  in  verticals  are  either  compression,  tension,  or  null. 
The  maximum  of  compressive  strain  occurs  when  the  diagonals 
in  connection  are  under  the  greatest  strain;  that  is,  under  an 
unequally  distributed  load.  For  other  explanation,  see  diagram 
under  variously-disposed  loads. 

In  the  following  formulas  and  examples  the  diagonals  (for  a 
moving  load)  resist  a  tensional  strain  only,  and  the  verticals  a 
compressive.  This  would  not  be  the  case  if  one  diagonal  to  each 
bay  were  used.  In  the  latter  case  the  diagonals  and  verticals 
would  have  to  resist  an  alternate  compressive  and  tensional  strain. 
When  the  trusses  are  inverted,  the  strains  are  different  in  kind, 
but  not  in  amount. 

Reference. 
A,  B  =  Reaction  of  support. 

(7=  Compression  in  arch  or  upper  boom. 
T—  Tension  in  lower  boom. 
D  and  H '=  Rise  of  arch. 
F  and  /  =  Vertical  forces. 

W=  Weight  of  moving  and  static  load  per  unit  of  span 

or  length. 

V=  Strain  in  verticals. 
N  =  Total  number  of  bays. 
a  =  Length  of  a  bay. 
c  =  Length  of  a  diagonal. 
d  and  h  =  Ordi nates  to  parabola. 

I  =  Distance  between  supports  or  span. 
k  =  Total  number  of  verticals  =  N —  1. 
m  =  Number  of  bays  between  support  and  Fn. 


148  STRAINS  IN  PAEABOLIC  CURVED  TRUSSES. 

n  =  Number  of  a  member,  counting  from  support  to 

middle  of  truss. 
t  =  Tension  in  diagonal. 

v  and  z  =  Angle  between  horizontal  and  member  of  polygon. 
w  =  Weight  of  static  load  per  unit  of  span  or  length. 
w/=  Weight  of  moving  load,  equally  distributed  per 

unit  of  span  or  length. 
u,  x,  y  =  Abscissas. 

In  the  following  diagrams,  one-half  of  truss  only  is  shown,  the 
strains  being  alike  in  the  respective  members  of  each  half: 


Fig.  229. 


Lower  Boom  Horizontal. 
To  find  the  ordinates  h  when  H  is  given : 


The  value  of  T given,  to  find  h: 


Fig.  230. 


Lower  Boom  Curved. 
To  find  the  ordinates  h  or  d  when  H  or  D  is  given: 

'  12  d»=  ~       % 


STRAINS  IN  PAEABOLIC  CURVED   TRUSSES. 


14ST 


The  value  of  T  given,  to  find  h: 
W(l—a)xn 


w 

~m~ 


Load  equally  distributed — Static  Load.     (Figs.  231  and  232.) 
W  =  The  weight  of  construction  and  applied  load. 


Fig,  231. 


IF/2 


Wl* 


Lower  Boom  Loaded. 

,  Wl 


wl 


H 


=  (7       V=  —-  =  tension 


Loaded. 


F=null. 


Fig.  232. 


Upper  Boom  Loaded.     (C=T.) 
Wl2  Wl'2 


V  =  — —  =  tension. 


150 


STRAINS   ITS  PARABOLIC  CURVED   TRUSSES. 


Load  unequally  distributed  —  Moving  Load.     (Figs.  233  and  234.) 

(Strains  in  Booms,  same  as  for  Static  Load.) 

Fig.  233. 


w/l 
- 


Lower  Boom  Loaded. 

Fn=  ^n—  A=  compression. 


Boom  Loaded. 


TFZ 


T7 

Vn=  --  =  compression. 


.  234. 


n=  — r—  =  compression 


—  D) 


STRAINS  Iff   PARABOLIC  CURVED   TRUSSES.  151 

EXAMPLE.    (Fig.  233.) 

Moving  Load  on  Lower  Boom. 

Reference. 

£  =  64  feet.  c1=  8.7  i'eet.  w  =  125  Ibs. 

H=  8  feet,  c2=  c3=  10.0  feet.  10,=  625  Ibs. 

a  =  8  feet,  c4=  c-=  10.9  feet.  W=w  +  w,=  750  Ibs. 

2V  =  8,  &  —  7.  c6=  11.3  feet. 

4  v  8  v  3(64  —  8^  ^i—     8.0  —   8  =      Ofeet. 

A1==       X      X--4^~—  -==  3.5  feet,  M2=    19.2  —  16=    3.2  feet. 

W3=    40.0  —  24  =  16.0  feet, 
^=  128.0  -32  =  96.0  feet, 


4X8X16(64-16) 

2==   "  2  -  = 


. 

A4=  IT  =8.0  feet. 
Tang.  t'1=  —^—  =  —  —  =  23° 


Tang.  v,=  h~^  =  ^—-^  =  3°  34'  30". 

y1=  3,5  x  2.28  =    8.0  feet.          ys=  7.5  X    5.37  =    40.0  feet. 
2/2=  6.0  X  3.20  =  19.2  feet.          y4=  8.0  X  16.00  =  128.0  feet. 


.  =  48,000  Ibs. 


(7n=  C  sec.  vn. 

Q=  48000  Xl090=52,3201bs.  C3=  48000  x  1.017  =48,816  Ibs. 
(72=48000x1-047=50,256^8.  C4=  48000  X  1.0019  =  48,091  Ibs. 

*=*%$%-*    8.7  =  5437.5  lb,        ,,- 
tf=  ts=  ^-^-  X  10.0  =  6250.0  Ibs. 


152  STRAINS   IN   PARABOLIC  CURVED   TRUSSES. 


625  x  64 

**=  **=   o^o   X  10.9  ==  6802.5  Ibs. 
°  X  o 

625  x  64 

^  —  ---—  X  11-3  =  7062.5  Ibs. 


=  2625 


2x  8 

I  =  1875 


=  1250 


j=  8(125+  625)--=  750 
2=  8  (125  +  625)  [-^yj-]  =  2250 
,=  8(125  -J  625) [J^t^L]  =  4500 
4=  8  (125  +  625)  I" JL-t^^_"j  =  7500 


PAKABOLIC  ARCHED   BEAMS  OR   BIBS.  153 


/4=   750  ( -- )=  562.5 

V  96  -f  4  X  8  / 

F1=6000    —0        =6,000  Ibs.   F3=  9000— 500    =8,500    Ibs. 
F2=  7812.5  — 312.5=  7,500 Ibs.   F4=  9375— 562.5  =  8,812.5  Ibs. 


CAPACITY  AND   STRENGTH   OF  PARABOLIC  ARCHED 
BEAMS  OR  RIBS  ORIGINALLY  CURVED. 

Reference.     (All  dimensions  in  inches.) 

A  =  Sectional  area  of  beam. 
C  =  Compressive  strain  in  direction  of  arch. 
E  =  Modulus  of  elasticity. 

H '  =  Horizontal  thrust  at  abutment,  or  tension  on  tie  rod. 
/=  Moment  of  inertia  of  cross-section  of  beam. 
R  =  Resistance  of  material  to  crushing,  (to  be  divided  by  factor 

of  safety.) 

W '  =  Concentrated  load  at  crown  of  arch. 
a  =  Vertical  deflection  at  crown. 
b  =  Horizontal  deflection  at  abutments. 
h  =  Rise  of  arch. 

21  =  Distance  between  abutments  =  span. 
s  =  Distance  between  neutral  axis  and  farthest  edge  of  section. 
w  =  Load  per  unit  of  length,  equally  distributed  horizontally. 
x  =  Vertical  distance  from  crown  to  point  of  arch,  intersected 

by  y,  say  at  0  on  diagram. 
y  =  Horizontal  distance  .from  middle  of  arch  to  section  where 

the  amount  of  strain  is  desired. 
v  =  Angle  between  horizontal  and  tangent  to  curve. 

Horizontal  Thrust,  (resisted  either  by  abutments  or  tie  rod.) 
Fig.  235.     (All  dimensions  to  line  of  pressure.) 


154  STKAINS   IN   A   POLYGONAL   FRAME. 

To  determine  the  curve  or  line  of  pressure: 
x  7/2  y 

—  =|r      -f 

2x         2<//^T~ 
Tang,  v  at  any  point  =  -  =  —  ^  — 

y          i 

2h 

Tang,  v  at  abutment  =  ^— 
t 

.  Load  concentrated  at  crown  or  middle  of  arch: 


,_25Z_          A  fty          25%2 

"  V  64/i  5(5Z  n      ^  32Z3    / 

__   25^     IF         81  PF/s 

**  —  "cTT      A  T 


1600J 
25Z  X  1600J 


64A(£  16007— 
Load  equally  distributed: 


_+_  /  . 

27*      p  ;^          l- 


STRAINS  IN  A  POLYGONAL  FRAME  IN  EQUILIBRIUM. 

Load  equally  distributed  over  members  of  Frame. 

Reference. 

H  =  Horizontal  strain  in  units  of  weight  at  foot. 
Fn=  Vertical  strain  in  units  of  weight  at  foot. 
(7n=  Compressive  strain  in  units  of  weight  in  direction  of 

member. 

TFn=  Load  in  units  of  weight,  equally  distributed  over  a  mem 
ber  of  the  polygon. 
fln=  Angle  between  horizontal  and  member. 


PARABOLIC   ARCHED   BEAMS  OR   RIBS.  155 

Fig.  236. 


H—  J  TFcotg.  vn  (7n=  Fn  cosec.  va 


2  2 

IT.+ 


,        3     _     ,  .         . 

1  s—  '  ^  •+  2  z   2  2 

_TFi    , 
" 


2 
W. 


2 
For  the  equilibrium,  vl  being  given : 


Tang.  v4=  -^r  =  tang.  ^ 
.a 


2+  TF3)  + 


H 

The  above  can  be  used  to  compute  the  strains  in  ribs  for  dome 
construction. 


156  STRAINS  IN  ROOF  TRUSSES. 


STRAINS  IN  ROOF  TRUSSES. 


Reference.     (Figs.  237  to  255.) 
C  Weight  of  construction.  } 

TF  =  <  Pressure  of  wind.  >—  Load  in  units  of 

(  Pressure  of  snow.  J 

weight,  equally  distributed  over  one  rafter. 
(See  Fig.  238.) 

C=  Compression  of  member  in  units  of  weight. 
T—  Tension  of  member  in  units  of  weight. 
L  =  Total  span,  or  distance  between   abutments  in 

units  of  length. 

d,  h,  I,  and  S  =  Dimensions  in  units  of  length.     (See  Figures.) 
v,  y  —  Angles.     (See  Figures.) 

The  diagrams  show  only  one-half  of  truss,  (except  Fig.  238.) 
the  thick  lines  indicating  compression,  and  the  thin  ones  tension. 
(See  "Reaction  of  Supports  "  for  pressure  on  joints  ;  also  "  Compound 
Strains  in  Trussed  Beams") 

Compression  in  Rafters.     (Trusses  Nos.  1,  3,  and  4.) 

The  compressive  strain  in  the  rafter  gradually  increases  from 
ridge  to  abutments.  Let  x  =  Horizontal  distance  from  abutment 

to  point  where  the  strain  is  desired,  and  I  half  the  span  =  -  . 


tg.  v 


C  for  Truss  No.  1  =  IF  sin.  v  (l — )  +  -HL 

Cfor  Truss  No.  3  =  IF  sin.  v(l  —  —}  +  —  — -°^ — 

1/2     tg.  (v  -|-  i\) 

C  for  Truss  No.  4  =  JFsin.  v(l  —  —}  +  —  ——- — 
V  I   n     2     ig.(v~Vl) 

In  the  following  examples  the  maximum  of  C  is  given : 

Truss  No.  1. 

Fig.  237. 

W    cos.  v 
C=W&m.' 


2      tg.t; 


W 

T=—  cotg.v 


STEAINS   Itf   ROOF   TRUSSES.  157 

EXAMPLE. 
Let  W—  8,000  Ibs. 
v  =  26°  30'. 

C=  8000  X  0.44619  +  -^  S§5T  =  10'666  lbs'    Com' 


7  8000 

When  a;  =  ~  then  will  C  =  -—  7^7777^  =  8,968  Ibs.  Com. 

A  2.    X    v^.TCTrOiy 

j_    80Q°   2.00  =  8,000  Ibs.     Tension. 


Truss  No.  2. 

Fig.  238. 


W  W 

T= sin.  v  cos.  v  =  — —  sin.  2v 

2  4 

EXAMPLE. 

Let  TT=  8,000  Ibs. 
v  =  26°  30'. 

8000 
C=  —  - —  X  0.4462  =  1,785  Ibs.     Compression. 

a 

C1=    8000    X  0.8952  =  6,568  Ibs.     Compression. 

8000 
T=  -^—  X  0.7986  =  1,597  Ibs.     Tension. 

[NOTE.— When  the  rafters  are  fastened  together  at  the  ridge,  they  are 
under  a  cross-breaking  strain  only.  Consequently  there  is  no  horizontal 
thrust  at  the  abutments ;  that  is,  T=0,  and  the  compression  in  <?i  ==  W.] 


158 


Fig.  239. 


STRAINS   IN    ROOF   TRUSSES. 

Truss  No.  3. 


n       TT7  .  W       cos. 

C  =  IF  sin.  v  -\ 

2    t.^- 


W         cos. 


2     sin.  (v  +  Vj) 


fy.  240. 


Truss  No.  4. 


0=  Ws'm.v 


W       cos.  v 


2 

cos.  v 


2     sin.(v  —  Vj) 

T,=  TF-^-~— ^~ 
sm.(v  —  vj 


Let  TF=  8,000  Ibs. 
v  =  26°  30X. 
v1==  5°  Ox. 

C  =  8000  X  0.44619  + 


EXAMPLE. 


=  12,653  Ibs.     Coin. 


?1  =  9,920  Ibs.     Tension. 


0.087 


O.ooo 


Bl|7201bB>   Tension. 


STRAINS  IN   HOOF   TRUSSES. 


159 


Truss  No.  5. 

Fig.  241. 


£=  ifTFcosec.  <y    Ci=  J  TP cotg.  v    T=  jl  +  -^--    TPcotg.  v 

When  there  is  no  tie  T,  (73  is  under  a  tensile  strain  ==  — n~, 

4/& 

7i  being  the  height  from  Cl  to  ridge. 

EXAMPLE. 
Let  TT=  8,000  Ibs. 
?  •=  22.36  feet. 
/  —  L==  11.18  feet. 

v  =  26°  3CK. 

C=  -}-|8000  X  2.241  —  14,566  Ibs.     Compression. 
Q—    %  8000  X  2.        =    8,OQO  Ibs.     Compression. 

T=  }  (l  +  —  — )  8000  X  2.  9 12,000  Ibs.     Tension. 
\          ^ii.oo  / 

Truss  No.  6. 

Fig.  242. 


160 


STEAINS   IN    ROOF   TRUSSES. 


,==  2(W—i\  W)  —  -- 


EXAMPLE. 
Let  TF=  8,000  Ibs. 
Z  =  20  feet. 
/1==  20.6  feet. 
h  =  10  feet. 
A!=  5  feet. 
£==  22.36. 

„        8000  x  500  —  1  500  (500  —  10  X  5) 

—  5X  2236   -        -^^  =  29,264  Ibs.     Com. 

Q=  0.625  x  SOOOfg  =  10,000  Ibs.     Compression. 
—  1500)  —JL  =  26,780  Ibs.     Tension. 

_   K 


Tj=  2(8000  —  1500) 


=  13,000  Ibs.     Tension. 


Truss  No.  7. 

Fig.  243. 


°  =  W~9J-'- 

M  sin.  v 
(?!=  if-  IF  cosec.  v 


Let  W=  8,000  Ibs. 
Z  =  20  feet. 


sin.  2v 


EXAMPLE. 

h  =  10  feet,  v  =  26°  30'. 

Zx=  11.18  feet.          vi==  26°  307. 


STRAINS   IN    EOOF   TRUSSES. 


161 


n  —  gQOO 


9Q 


- 
2  X  20  X  0.44619 


•  =  8,964  Ibs.     Compression. 


—  08125  X  8000  X  2.2411  =  14,567  Ibs.  Compression. 
2==  0.625  X  8000  X  1-12  =  5,600  Ibs.  Compression. 
'=  0.625  X  8000  =  5,000  Ibs.  Tension. 


(7 

G=: 

T=    .  ,  . 

2i=  0.8125  X  8000  X  2.0  =  13,000  Ibs. 


Truss  No.  8. 

Fig.  244. 


c  -=  .  =  w 

2sin.v  2^sin 


s'm.(v  — 


=  |  T7    2 
h 


sin-  ^     i 


cos,  vain.  fa—  i 


"1  _ 

J  ~ 


Let  TF^8,000  Ibs. 
v  =  26°  30/. 
t;!=  9°  20'. 
up=  19°  0'. 
11 


EXAMPLE. 


162  STRAINS  IN  ROOF  TRUSSES. 


9000  +  0.375  x  8000 

C=  -  -  =  13,452  Ibs.     Compression. 

0.892 


Ci—  0.812  X  8000  ~r  =  21,710  Ibs.     Compression. 
u.zyo 

0.2=  0.625  X  8000  A8^  =    7,535  Ibs.     Compression. 


T  =  0  812  X  8000  —  -~-  =  19,702  Ibs.     Tension. 
(j.2t  Jo 


Truss  No.  9, 

5i<7.  245. 


C=  if  fF_J: f  IF  sin.  v 

sin.  v 

1 

(;,—  j|  FF--^ =  -^fTFcosec.  v 


rj\=  |f  17 cotg.  v  —  TV  IF cotg.  v  =  -}-  IF  cotg.  v 
T,=  i'IWcotg.v 

EXAMPLE. 
Let  TF^  8,000  Ibs. 

v  =  26°  307. 

C  =  0.812x  8000  x  2.241  — 0.625  X  8000  X  0.4 16  =  12,336  Ibs. 
Compression. 

Ci=  0.812  X  8000  X  2.241  =  14,56G  Ibs.     Compression. 

C2—  0.625  X  8000  X  0.895  —    4,475  Ibs.     Compression. 

T—  0.312  X  8000  X  2.  =  4,992  Ibs.     Tension. 

2\=  0.812  X  8000  X  2— 0.312X  8000X  2.  =  8,0001bs.  Tension. 

T2=  0.812  X  8000  X  2.  =  12,992  Ibs.     Tension. 


STRAINS   IN   ROOF   TRUSSES.  163 

Truss  No.  1O. 

Fig.  246. 


~~N I  IF  sin.  v 


<72=f  JFcos.v. 


T_    COS.  V  COS.  Vi 

2\=  if  W^^--~ Tcos.  (2v  —  vi)  —  -I  Wain.  cos.  v 

W       I 


EXAMPLE. 

Let  TF=  8,000  Ibs.  Vl=  9°  20^  A  =  10  feet 

v  =  26°30/.  1  =  20  feet.  A,=    2  feet. 


(7=0.8125  X  8000  --  0.625  x  8000  X  0.446  =  19,517  Ibs. 

Compression. 

0  987 
(7i=  0.8125  X  S000^1^-  ==  21.747  Ibs.     Compression. 

OF=  0.625    x  8000  x  0.895  =  4,475  Ibs.     Compression 

o,25  X  3000  x 


0.8952]  =  7,163  Ibs.     Tension. 


164  STRAINS   IN   ROOF   TRUSSES. 


Tl=  _  X  -j-  =  10,000  Ibs.    Tension. 


T»=  0.8125  X  8000  =  19,720  Ibs.     Tension. 


Truss  No.  11. 

Fig.  247. 


(7=13  IF    .  C°S-  Vl     --  -  f  TF  sin.  * 
— 


cos.  y 
T=^W—  (-1/ —  - [-5  cos.  v\ 

~W       I  cos.  v 

2    h  —  hi  sin.^ —  Vi) 

EXAMPLE. 

Let  W=  8,000  Ibs.    y  ==  50°.        h  =  10  feet.      I  =  20  feet. 
v  =  26°30/.       w1=9°20/.    Ai=2feet.       S  =  22.36  feet. 

(7=  0.8325X8000  ^'^   —0.625  X  8000  X  0.446  =  19,517  Ibs. 
Compression. 


=  0.8125  X  8000  ~-  =  21,747  Ibs.    Compression, 
u.^yo 

=  0.366  x  8000  %~.  =   4,070  Ibs.    Compression. 


STRAINS  IN  EOOF  TRUSSES. 


165 


2-  =  0.125  X  8000  ™       (6.5  .  +  5  .  0.894)  =  11,050  Ibs. 

Tension. 

7i=  19486  X  0.986  —  7421  x  0.723  —  4930  X  0.446  =  10,000  Iba. 
Tension. 

0  S94 

Tf=  0.812  X  8000  -^-^  =  19,486  Ibs.  Tension. 
0.29o 


Truss  No.  12. 

Fig.  248. 


Let  P7=  8000  Ibs. 
Z  =  20  feet. 


EXAMPLE. 


A  =  10  feet. 
/S'  =  22.36  feet. 


20 
C71=  0.366  X  8000  —  =  5,856  Ibs.    Compression. 

8000    22  sr 

(72=  0.366  X  --  5—  —  ~  =  3,280  Ibs.     Compression. 


10 


=  0.866  X 


10 


20,992  Ibs.    Tension. 


3;=  1.23    X  8000  =  9,840  Ibs.     Tension. 


166 


STRAINS   IN   EOOF   TRUSSES. 


Truss  No.  13. 

Fig.  249. 


?i=«*T- 


cos.  vt 

C,^  f|  W : -. 

-  -1  J  01  r>     It,  


e4=  ft  x 


sin.(D  — «L 

A. 

•A. 

k 


T-  Wk        *  W 
2 —  ~T i?  w 


EXAMPLE. 

Let  TF=  20,000  Ibs.  h  =  20     feet,          v  =  21°  40X. 

2  =  50  feet.  12=  53.8  feet.          v  =  0°. 

CO    O 

C  =  0.5  X  20000  -— —  =  26,900  Ibs.  Compression. 
(71=  0.683  X  20000  — ~  =  37,018  Ibs.  Compression. 
C2=  0.866  X  20000  *  =  46,937  Ibs.  Compression. 


(73=0.55    x  20000  -^-i-  =  11,770  Ibs.     Compression. 


Q=0.55    x  20000  4^  =9,900  Ibs.    Compression. 


STRAINS   IN    EOOF   TRUSSES. 


167 


T=  0.683  X  20000  X  2.517  =  34,382  Ibs.     Tension. 
T^=  0.866  X  20000  X  2.517  =  43,594  Ibs.     Tension. 

20000  X  20  =  14)666  lbg>     Tensiont 


20 
T3=  0.183  X  20000  =  3,660  Ibs.     Tension. 

Truss  No.  14. 

Fig.  250. 


h          °        h 

rp n     ^2 

J-b —  O6— T 


EXAMPLE. 

Let  TF=  24,000  Ibs.     Span  =  100  feet    1=  Z1=  Z2=  ?3=  1.25  feet. 
A  =  20  feet.  JJ^O.  ^^53.85  feet. 


168 


STRAINS   IN   ROOF   TRUSSES. 


53  85 
Q=  12000  x  — ~  =  32,310  Ibs.     Compression. 

<72=  49088  -  0.228  X  24000  ^L  =  41,728  Ibs.     Com. 

C8=  58320  —  0.286  x  24000  ^®L  =  49^88  Ibs.     Com. 
^  X  20 

£4=  21600  -^0--  =  58,320  Ibs.     Compression. 

C5=  (5801  +  0.286  X  24000)  ~^--  =  12,493  Ibs.     Com. 

Q=  3432  +  5484  ~~-  =  9,282  Ibs.     Compression. 

13  47 
0,=  0.286  X  24000  ~^~  =  9,245  Ibs.     Compression. 

T,=  (24000  —  0.1  x  24000)  -^-  =  54,000  Ibs.     Tension. 
7!,=  5 1000  —  0.286  x  21000  -~-  =  45,420  Ibs.     Tension. 

7!,=  45120  —  9282  --f\—  38>170  Ibs.     Tension. 
In 

7^=24000  —  -121000=  19,200  Ibs.     Tension. 

r5=  9282  J£-  =  5,801  Ibs.     Tension, 
lo 

TG=  0.286  x  24000  ~  =  3,432  Ibs.     Tension. 


Truss  No.  15. 

Fig.  251. 


y  i  F 


STRAINS   IN   ROOF   TRUSSES.  169 


C=  if  W  ---     L^_^  --  £|  TFsin.  v  —  JJ  TFcos.  v  cotg.  (v  —  vj 


__ 
sin.  (v  —  ^)  2    (A  — 


-—-.-—  tang.  yi  T5=  \$W     .        '        , 

(h  -Ax  )  sm.('y  — -yj) 


EXAMPLE. 

Let  W=  20,000  lbs.         h  =  20  feet.          v1==  0. 

I  =  50  feet,  v  =  21°  40'.         -v2=  46°  30'. 

tf  =  0.866  X  20000      *       —  0.733  X  20000  X  0.369  —  0.183  X 
0.369 

20000  X  0.929  x  2.517  =  32,959  lbs.     Compression. 

C1==  0.866  X  20000  X  -A~T^ 0.366   X    20000   x   0.369 

0.369 

=  44,236  lbs.     Compression. 
C.2=  0.866  X  20000  X  7^7-  =  46,937  lbs.     Compression. 


Q=  0.55    x  20000  X  0.929  =  10,219  Ibs.     Compression. 
C4=  0.366  X  20000  X  0.929  =    6,800  Ibs.     Compression. 

40 
T=  20000  X  -T^-  X  tang,  v  =  Null. 


=  10,9201bs.     TenSIOn. 
Tf=  0.183  X  20000  X  2.5  =  9,150  Ibs.     Tension. 
TZ=  10000  X  -2Q^°—  =  25-00°  lbs-     Tension. 

T,=  0.683  X  20000  X  2.5  =  34,150  lbs.     Tension. 
2^=  0.866  X  20000  X  2.5  =  43,300  lbs.     Tension. 


170 


STRAINS   IN    ROOF    TRUSSES. 

Truss  No.  16. 

Fig.  252. 


C'4=f  JFcos.v. 

Q=  j|>  JFcos.  v  +  f-  IF  cos.  v  =  JfTFcos.  v 


—- 
sm.(2v— 


W_     __l_ 
~T    h  — 


'F  9    TJ7" 

^4=  TO   W~Z 


sin.^  —  Vi) 
cos.  v 


sin.  (v  — 


EXAMPLE. 

Let  TT=  20,000  Ibs.  A  =  20  feet.  AI=  0. 

I  =  50  feet.  T;  =  21°  407.  vi.=  0. 

0=  41885  — 0.286  X  20000  x  0.369  =  39,774  Ibs.  Compression. 
Cj=  43567  —  0.228  X  20000  X  0.369  =  41, 885  Ibs.  Compression. 


STEAINS   IN    ROOF    TRUSSES.  171 

<72=  48780  —  5213  =  43;567  Ibs.     Compression. 
(73=  0.9  X  20000 —  =  48,780  Ibs.     Compression. 

<?==  0.286  X  20000  X  0.929  =  5.213  Ibs.     Compression. 
Q=  0.514  X  20000  X  0.929  =  9,550  Ibs.     Compression. 

T==  ( 0.9  X  20000  -~— -  —  0.8  X  20000  X  0  369*  —  0.1  X 
\  0.369 

20000)  — * —  =  20,000  Ibs.     Tension. 
/   0.686 

2i=  T—  T5=  20000  —  7188  =  12,812  Ibs.     Tension. 


_ 

2  20 

0  9°9 

T3=  T±—  T5=  0.757  X  20000  —     —  =  38,1  18  Ibs.     Tension. 

U.oo  3 

0  D99 
T±=  0.9  X  20000  -       -  =  45,306  Ibs.     Tension. 


f=  Tfi=  T—  Tl=  7,188  Ibs.     Tension. 
6=  T^=  7,188  Ibs.     Tension. 

Truss  No.  17. 

Fig.  253. 


When  the^  rafter  is  resting  on  joint  A: 

n  _        W  _       TFcos.  v  cos.^  —  -y) 

0  --  -  —  r  -  —  Go  —  -%  --  ;  -  -  - 

4  sin.  v  sm.  vl 

W 

C?1=-r-T  -  r 
4  sin.-v 

„      ,     TT  cos. 


l=  0,  cos. 


sm.  ^ 
Bending  moment  at  point  B  —  C!2  sin.  v,  .  J. 


172  STRAINS  IN   ROOF   TRUSSES. 

When  rafter  is  fixed  at  joint  A: 


_ 

C  - 


W 

-  r  - 

4  sin.  v 


_       W  cos.  v  cos.  (i\  —  v) 

Go  -  %  -  :  -- 

sm.  v1 
T=  JPFcotg.  VI+TI 

T}=  ^-  cotg.  v 


IF 
Bending  moment  at  B  =  —  -  . 

£ 


Truss  No.  18. 

Fig.  254. 


TFcos. 


ri  —  i 

T=Q 

Ti=  C3  cos.  v  4-  C2  cos. 


STRAINS   IN   ROOF    TRUSSES. 


173 


Truss  No.  19. 

Fig.  255. 


C  =    \W cosec.  v 
.  v 
C2=  Jf  W  cosec.  v 

a=  §  w  cotg.  v 


^i=  f  TFcotg.  v  +  JTF  tang, 
T2=  flF  cotg.  v 


EXAMPLE. 
Let  W=  20,000  Ibs.         v  =  21°  40'.         t>1==  56° 


C—  27,000  Ibs. 
C}=  36,900  Ibs. 
<72=  46,800  Ibs. 
C3=  33,466  Ibs. 
Q=    6,867  Ibs. 

Compression. 
Compression. 
Compression. 
Compressian. 
Compression. 

(75=    3,533  Ibs. 
(76=    6,666  Ibs. 
T=    6,666  Ibs. 
7\=  37,000  Ibs. 
r2=  41,831  Ibs. 

Compression. 
Compression. 
Tension. 
Tension. 
Tension. 

174 


STRAINS  IN  ROOF  TRUSSES. 


M 

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II  8? 
«>S 


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s  £ 


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; 

6  ^o 
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STRAINS   IN   EOOF   TRUSSES. 


175 


O  1O  OC  tO 
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rH   r-l   rH  CD 

CM  CM  tC  CM 

CO   r-H  01  CO 
rH  CO  CO  O 

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CO   r-H  CD  O   rH 
rH  CO  1C   1C   rH 

CO  CM  rH  1C  CO 
CM  iC  ^H  1^-  CO 

r^  oo  co  co  oq 

oo  ro  oo  T—  i 

II  II  II  II 

0^5^^ 

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II      II      II       II      II 

o^cS^s^ 

rh  CM  O  1C 
(M  ^H  01  01 

1C  OC'  CO  CO 

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1C  'xH  Ol  LQ 
1.^  *5f  CO  CO 

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STRAINS  IN  ROOF  TRUSSES. 


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178 


PRESSURE   OF   WIND   ON    ROOFS. 


EXAMPLE  TO  TABLE  OF  CONSTANTS.     (Tniss  No.  13.) 

What  is  the  amount  of  strain  in  the  various  members  of  a  truss, 
according  to  Fig.  249,  of  the  following  dimensions,  viz:  Span  60 
feet,  distance  between  trusses  10  feet,  height  at  center  10  feet, 
weight  to  be  carried,  including  weight  of  construction,  66|  Ibs. 
per  square  foot  horizontally;  hence  total  weight  on  one  rafter 
=  30  X  10  X  66}  =  20,000  Ibs.? 

L  =  60  feet.  r  6Q  v  =  18°  20'. 

h  =  10  feet.          —  =  —  =  6.          W  =  20,000  Ibs. 


^ 

o" 


Member.  Constant.        W  Strains. 

C2  =  2.745  x  20,000  =  54,900  Ibs. 
C,  =  0.660  X  20,000  =  13,200  Ibs.  J.  Compression. 
C±  =  0.567  X  20,000  =  11,340  Ibs.  } 
T  =  1.956  x  20,000  =  39,120  Ibs. ' 
T.  =  2.606  x  20,000  ==  52,120  1 
K  =  0.734  x  20,000  =  14^680  Ibs.  , 
Ta  «  0.183  X  20,000  =    3,660  Ibs.  J 


Tension. 


[NOTE.— la  the  foregoing  table  the  proportion  of  7i  to  L  is  approximate. 
The  constants  are  based  on  the  angles.] 


PRESSURE  OF  WIND  ON  ROOFS. 

In  the  following  table  the  maximum  pressure  of  wind  is  taken 
at  50  Ibs.  per  square  foot: 

The  angle  between  horizontal  and  direction  of  wind  is  generally 
10°  00r.     (See  diagram.) 

Fig.  256. 


Reference. 

F  =  Force  of  wind  in  Ibs.  =  50. 

w,  =  Pressure  at  right  angles  to  surface  per  square  foot  in  Ibs. 

w//=  Pressure,  vertical,  per  square  foot  in  Ibs. 

w/ =  F  sin.2  (v  +  10) 

w, 
w^= 

COS.  V 


PRESSURE  OF  WIND   ON  ROOFS. 


Proportion  of 
height  h  to 
span  I. 

Angle  v. 

Pressure  w, 
in  Ibs. 

Pressure  wtl 
in  Ibs. 

££ 

90°  00' 

50.00 

0.00 

*4 

45°  00' 

33.53 

47.40 

&=4- 

33°  41'  50" 

23.80 

28.60 

z 

26°  33'  50" 

17.64 

19.70 

*=-g- 

21°  48' 

13.83 

14.80 

;t=4 

18°  26' 

11.23 

11.80 

z 

"7 

15°  54'  40" 

9.46 

9.80 

A=-L 

14°  02'  10" 

8.56 

8.80 

*=-r 

12°  31'  40" 

7.29 

7.40 

•i-i 

11°  18'  40" 

6.51 

6.60 

180 


PEESSUEE  OF  SNOW  ON  EOOFS. 


PRESSURE  OF  SNOW  ON  ROOFS. 

The  average  pressure  of  snow  on  a  level  surface,  in  the  United 
States,  is  about  15  Ibs.  per  square  foot. 

The  following  table  gives  the  pressure  per  square  foot  on 
various  inclined  surfaces : 

Reference. 

P  =  Pressure  per  square  foot  in  Ibs.  =  15. 

p1=  Vertical  pressure  in  Ibs. 

pz=  Pressure  at  right  angles  to  surface  in  Ibs. 

v  =  Angle  between  surface  and  horizontal. 

p1=  P  cos.  v. 


Proportion  of 
height  h  to 
span  I. 

Angle  v. 

Pressure  PI 
in  Ibs. 

Pressure  P2 
in  Ibs. 

I 
~2~ 

45°  00' 

10.60 

7.49 

'--r 

33°  41'  50" 

12.48 

10.38 

*—  T 

26°  33'  50" 

13.42 

12.00 

*=4- 

21°  48' 

13.93 

12.94 

*=4- 

18°  26' 

14.23 

13,50 

*=4 

15°  54'  40" 

14.41 

13.86 

*=i 

14°  02'  10" 

14.52 

.  14.05 

*=4- 

12°  31'  40" 

14.64 

14.29 

h==Jo 

11°  18'  40" 

14.71 

14.43 

I 

0°  00'  00" 

15.00 

15.00 

TIE  RODS   AND   BARS. 


TIE  BODS  AND  BARS. 

Capacity  and  Proportional  Dimensions  of  Wrought-iron  Tie  Rods 
Tie  Bars,  and  Pins  or  Bolts. 

Ultimate  resistance  to  tearing  =  60,000  Ibs.  =  30  tons  pei 
square  inch. 

Ultimate  resistance  to  shearing  =  50,000  Ibs.  =  25  tons  pe: 
square  inch.  (See  Fig.  258.) 


Capacity  of  tie  or  bar. 

t 
d 

inches, 
d. 

Dimension  of 
flat  bars  in  in., 
uniform  thick 
ness. 

Diamete 
D  of  pii 
or  bolt. 

2  1 

d   C 

en 

«4 

o> 

O  SJO 

CO    b 

3  times  safety. 

5  times  safety. 

"ol  C 

g.rH 
O 

**  2 
Q.,-1 

Jj! 

0 
rO 

§0 

O   p 

ci  £ 

Lbs. 

Tons. 

Lbs. 

Tons. 

1 

CO 

1 
3 

I- 

I"8 

OJ 

a  cc 

PH  c 
C  X 
-^ 

5,000 

2.50 

3,000 

1.50 

0.25 

0.56 

¥ 

1 

0.75 

0.62 

0.4 

6.200 

3.10 

3.720 

1.86 

0.31 

0.62 

\\/ 

0-93 

0.69 

0.4 

7,400 

3.70 

4,440 

2.22 

0.37 

0.70 

<< 

11A 

1.12 

0.75 

0.5 

8,000 

4.30 

5,160 

2.58 

0.43 

0.74 

" 

1^4 

1.31 

0.80 

0.5 

10,000 

5.00 

6,000 

3.00 

0.50 

0.79 

" 

2 

1.50 

0.88 

0.0 

11.200 

5.00 

6,720 

3.36 

0.56 

0.84 

" 

2i/ 

1.68 

0.92 

0.0 

12,400 

6.2D 

7,440 

3.72 

0.62 

0.89 

«« 

2\s 

1.87 

0.97 

o.o 

13,000 

6.80 

8.160 

3.88 

0.68 

0.93 

" 

2^4 

2.06 

1.01 

0.7 

15,000 

7.50 

9,000 

4.50 

0.75 

0.97 

" 

3 

2.25 

1.08 

0.7 

7,400 

3.70 

4,440 

2.22 

0.37 

0.68 

% 

1 

0.75 

0.75 

0.6 

9,200 

4.60 

5,520 

2.76 

0.40 

0.76 

« 

\\s 

0.93 

0.83 

O.o 

11,200 

5.00 

6,720 

3.36 

0.56 

0.84 

" 

l1^ 

1.12 

0.92 

0.0 

13,000 

0.50 

7,800 

3.90 

0.65 

0.91 

" 

]^ 

1.31 

0.99 

0.7 

15,000 

7.50 

9,000 

4.50 

0.75 

0.97 

" 

2 

1.50 

1.08 

0.7 

10,800 

8.40 

10,080 

5.04 

0.84 

1.04 

" 

2/4 

1.68 

1.13 

0.8 

18,000 

9.30 

11,100 

5.58 

0.93 

1.09 

" 

2/^ 

1.87 

1.19 

0.8 

20,000 

10.30 

12,300 

6.18 

1.03 

1.15 

" 

2^4 

2.06 

1.24 

0.8 

22,400 

11.20 

13,440 

6.72 

1.12 

1.19 

" 

3 

2.25 

1.29 

0.9 

10,000 

5.00 

6,000 

3.00 

0.50 

0.79 

1A 

1 

0.75 

0.88 

0.0 

12,400 

6.20 

7,440 

3.72 

0.02 

0.88 

tf 

0.93 

0.97 

0.0 

15,000 

7.50 

9,000 

4.50 

0.75 

0.97 

« 

|i/ 

1.12 

1.08 

0.7 

17,400 

8.70 

10,440 

5.02 

0.87 

1.05 

M 

l^i 

1.31 

1.16 

0.8 

20,000 

10.00 

12,000 

6.00 

1.00 

1.13 

" 

2 

1.50 

1.24 

O.S 

22,400 

11.20 

13,440 

6.72 

1.12 

1.20 

" 

214 

1.68 

1.32 

0.9 

25,000 

12.50 

15,000 

7.50 

1.25 

1.26 

" 

2M 

1.87 

1.39 

0.9 

27.400 

13.70 

16,440 

8.22 

1.37 

1.32 

" 

2% 

2.00 

1.45 

1.0, 

30^000 

15.00 

18,000 

9.00 

1.50 

1.39 

" 

3 

225 

1.52 

1.0 

12,400 

6.20 

7,440 

3.72 

0.62 

0.90 

% 

1 

0.75 

0.98 

0.0' 

15.600 

7.80 

9,300 

4.68 

0.78 

1.00 

" 

\\s 

0.93 

1.09 

0.7' 

18,600 

9.30 

11,160 

5.58 

0.93 

1.09 

*• 

\\£ 

1.12 

1.20 

0.8, 

21,800 

10.90 

13,080 

6.54 

1.09 

1.18 

M 

]%/ 

1.31 

1.29 

0.91 

25,000 

12.50 

15,000 

7.50 

1.25 

1.26 

M 

2 

1.50 

1.39 

0.9! 

28,000 

14.00 

16,800 

8.40 

1.40 

1.34 

" 

2/4 

1.08 

1.47 

1.0 

30,533 

15.27 

18,720 

9.36 

1.56 

1.41 

" 

2V 

1.87 

1.54 

l.OJ 

TIE   RODS  AND  BARS. 


Capacity  of  tie  or  bar. 

£ 
®  g 

« 

0 
ft 

a 

c  . 

rt  C 

Dimension   of 
flat  bars  in  in., 
uniform  thick 
ness. 

Diameter 
D  of  pin 
or  bolt. 

7.  jn 

.5  o 

GO 

• 

• 

*  bb 

3  times  safety. 

5  times  safety. 

ol  C 

§" 

fl 

V     «H 

*'.: 

*  c 

|| 

If 

Lbs. 

Tons. 

Lbs. 

Tons. 

I 

s 

li 

H 

t* 

^  c 

ol 

l| 

§1 

34,200 

17.10 

20,520 

10.26 

1.71 

1.48 

~% 

2% 

2.06 

1.62 

1.14 

37,500 

18.75 

22,440 

11.22 

1.87 

1.54 

"8 

2.25 

1.69 

1.20 

15,000 

7.50 

9,000 

4.50 

0.75 

0.98 

% 

1 

0.75 

1.08 

0.76 

18,600 

9.30 

11,160 

5.58 

0.93 

1.09 

M 

0.93 

1.20 

0.85 

22,400 

11.20 

13,440 

6.72 

1.12 

1.19 

" 

\\/ 

1.12 

1.31 

0.93 

26,200 

13.10 

15,720 

7.86 

1.31 

1.30 

" 

1§& 

1.31 

1.41 

1.00 

30,000 

15.00 

18,000 

9.00 

1.50 

1.39 

" 

2 

1.50 

1.52 

1.08 

33,600 

16.80 

20,160 

10.08 

1.68 

1.46 

" 

1.68 

1.62 

1.14 

37,400 

18.70 

22,440 

11.22 

1.87 

1.54 

" 

2V" 

1.87 

1.69 

1.20 

41,200 

20.60 

24,720 

12.36 

2.06 

1.62 

" 

2% 

2.06 

1.77 

1.26 

45,000 

22.50 

27,000 

13.50 

2.25 

1.69 

" 

3 

2.25 

1.86 

1.32 

17,400 

8.70 

10,440 

5.22 

0.87 

1.05 

% 

1 

0.75 

1.16 

0.82 

21,800 

10.90 

13,080 

6.54 

1.09 

1.18 

0.93 

1.29 

0.91 

26,200 

13.10 

15,720 

7.86 

1.31 

1.29 

• 

1|| 

1.12 

1.41 

1.00 

30,600 

15.30 

18,360 

9.18 

1.53 

1.40 

«« 

1.31 

1.53 

1.08 

34,800 

17.40 

20,880 

10.44 

1.74 

1.49 

" 

2  4 

1.50 

1.63 

1.16 

39,200 

19.60 

23,520 

11.76 

1.96 

1.58 

" 

2*4 

1.68 

1.73 

1.23 

43,600 

21.80 

26,160 

13.08 

2.18 

1.66 

11 

2/lz 

1.87 

1.82 

1.29 

48,000 

24.00 

28,800 

14.40 

2.40 

1.75 

" 

2% 

2.06 

1.89 

1.34 

52,400 

26.20 

31.440 

15.72 

2.62 

1.83 

" 

3 

2.25 

2.00 

1.42 

20,000 

10.00 

12,000 

6.00 

1.00 

1.13 

1 

1 

0.75 

1.39 

0.80 

25,000 

12.50 

15,000 

7.50 

1.25 

1.26 

" 

i/4 

0.93 

1.45 

0.98 

30,000 

15.00 

18,000 

9.00 

1.50 

1.39 

" 

\\z 

1.12 

1.52 

1.08 

35,000 

17.50 

21,000 

10.50 

1.75 

1.49 

" 

\3S 

1.31 

1.64 

1.16 

40,000 

20.00 

24,000 

12.00 

2.00 

1.60 

" 

2 

1.50 

1.75 

1.24 

45,000 

22.50 

27,000 

13  50 

2.25 

1.70 

" 

2/4 

1.68 

1.86 

1.32 

50,000 

25.00 

30,000 

15.00 

2.50 

1.79 

M 

2V^ 

1.87 

1.96 

1.39 

55,000 

27.50 

33,000 

16.50 

2.75 

1.87 

" 

2% 

2.06 

2.05 

1.45 

60,000 

30.00 

36,000 

18.00 

3.00 

1.96 

*' 

3 

2.25 

2.15 

1.52 

28,000 

14.00 

16,800 

8.40 

1.40 

1.34 

iy& 

VA 

0.93 

1.47 

1.04 

33,600 

16.80 

20,160 

10.08 

1.68 

1.47 

" 

i/^ 

1.12 

1.60 

1.13 

39,600 

19.80 

23,520 

11.76 

1.98 

1.58 

" 

i?^ 

1.31 

1.73 

1.23 

45,000 

22.50 

27,000 

13.50 

2.25 

1.69 

" 

2 

1.50 

1.86 

1.32 

50,600 

25.30 

30,360 

15.18 

2.53 

1.80 

" 

2/4 

1.68 

1.97 

1.39 

56,200 

28.10 

33,720 

16.86 

2.81 

1.89 

" 

2V^ 

1.87 

2.09 

1.48 

61,800 

30.90 

37,080 

18.54 

3.09 

1.98 

" 

2% 

2.06 

2.18 

1.54 

67,400 

33.70 

40,440 

20.22 

3.37 

2.08 

" 

3 

2.25 

2.26 

1.60 

73,000 

36.50 

43,800 

21.90 

3.65 

2.16 

* 

3/4 

2.43 

2.36 

1.67 

78,600 

39.30 

47,160 

23.58 

3.93 

2.24 

" 

3/^j 

2.62 

2.45 

1.74 

84,200 

42.10 

50,520 

25.26 

4.21 

2.32 

* 

3^4 

281 

2.53 

1.80 

90,000 

45.00 

54,000 

27.00 

4.50 

2.40 

• 

4 

3.00 

2.63 

1.86 

31,200 

15.60 

18,720 

9.36 

1.56 

1.41 

li,/ 

|i/ 

0.93 

1.54 

1.09 

37,400 

18.70 

22,440 

11.22 

1.87 

1.55 

w 

li^ 

1.12 

1.69 

1.20 

43,600 

21.80 

26,160 

13.08 

2.18 

1.67 

• 

1% 

1.31 

1.82 

1.29 

50,000 

25.00 

30,000 

15.00 

2.50 

1.79 

• 

2 

1.50 

1.96 

1.39 

56,200 

28.10 

33,720 

16.86 

2.81 

1.89 

(t 

2/4 

1.68 

2.09 

1.48 

62,400 

31.20  , 

37,440 

18.72 

3J2 

1.99 

" 

2/^ 

1.87 

2.19 

1.55 

TIE   HODS   AND    BARS. 


Capacity  of  tie  or  bar. 

cr 

CO 

(J 

03  ^ 

'o 

a   - 

Dimension  of 
flat  bars  in  in., 
uniform  thick 
ness. 

Diameter 
D  of  pin 
or  bolt. 

3  times  safety. 

5  times  safety. 

S'o 

1! 

00 

tn     . 
0>  ~ 

*>: 

*! 

§t 

if 

O 

<£>•-. 
1 

•g£ 

|£ 

"2  § 

si 

iis 

O  J3 

Lbs. 

Tons. 

Lbs. 

Tons. 

CB 

Q 

£~ 

g° 

£2 

03 

G  co 

0=3 

£_,«*- 

^  0 

68,600 

3430 

41,160 

20.58 

3.43 

2.10 

il/4 

2% 

2.06 

2.29 

1.62 

75,000 

37.50 

45,000 

22.50 

3.75 

2.19 

" 

3 

2.25 

2.40 

1.70 

81,200 

40.60 

48,720 

24.36 

4.06 

2.27 

« 

3/4 

2.43 

2.49 

1.7G 

87,400 

43.70 

52,440 

26.22 

4,37 

2.36 

" 

31^ 

2.62 

2.60 

1.84 

93,600 

46.80 

56,160 

28.08 

4.68 

2.44 

" 

3% 

2.81 

2.68 

1.89 

100,000 

50.00 

60,000 

30.00 

5.00 

2.53 

" 

4 

3.00 

2.77 

1.96 

41,200 

20.60 

24,720 

12.36 

2.06 

1.62 

Jg 

^A 

1.12 

1.77 

1.26 

48,000 

24.00 

28,800 

14.40 

2.40 

1.75 

" 

1/4 

1.31 

1.89 

1.34 

55.000 

27.50 

33,000 

16.50 

2.75 

1.87 

" 

2 

1.50 

2.05 

1.45 

61,800 

30.90 

37,080 

18.54 

3.09 

1.98 

« 

2// 

1.68 

2.18 

1.54 

68,600 

31.30 

41,160 

20.58 

3.43 

2.09 

<« 

2/<2 

1.87 

2.29 

1.62 

75,600 

37.80 

45.360 

22.68 

3.78 

2.19 

-% 

2.06 

2.41 

1.71 

82,400 

41.20 

49,440 

24.72 

4.12 

2.29 

3 

2.25 

2.51 

1.78 

89,200 

44.60 

53,520 

26.76 

4.46 

2.38 

3/4 

243 

261 

1.85 

96,200 

4810 

57,720 

28.86 

4.81 

2.47 

3^2 

2.62 

2.71 

1.92 

103,000 

51.50 

61,800 

30.90 

5.15 

2.56 

3% 

2.81 

2.81 

1.99 

110,000 

55.00 

66,000 

33.00 

5.50 

2.65 

4 

3.00 

2.90 

2.05 

45,000 

22.5 

27,000 

13.50 

2.25 

1.70 

1A 

\y 

1.12 

1.86 

1.32 

52,400 

26.20 

31.440 

15.72 

2.62 

1.83 

1^4 

1.31 

2.00 

1.42 

60,000 

30.00 

36,000 

18.00 

3.00 

1.96 

2 

1.50 

2.15 

1.52 

67,400 

33.70 

40,440 

20.22 

3.37 

2.07 

2/4 

1.68 

2.27 

1.61 

75,000 

37.50 

45,000 

22.50 

3.75 

2.19 

2Vo 

1.87 

2.40 

1.70 

82,400 

41.20 

49.440 

24.72 

4.12 

2.29 

2% 

2.06 

2.51 

1.78 

90,000 

45.00 

54,000 

27.00 

4.50 

2.40 

3 

2.25 

2.63 

1.86 

97,400 

48.70 

58.440 

29.22 

4.87 

2.49 

3/4 

2.43 

2.73 

1.93 

105,000 

52.50 

63,000 

31.50 

5.25 

2.59 

3/^2 

2.62 

2.84 

2.01 

113,400 

56.20 

67,440 

33.72 

5.62 

2.67 

3% 

2.81 

2.93 

2.08 

120,000 

60.00 

72,000 

36.00 

6.00 

2.77 

4 

3.00 

3.03 

2.15 

127,400 

63.70 

76.440 

38.22 

6.37 

2.85 

4/4 

3.18 

3.12 

2.21 

135  000 

67.50 

81,000 

40.50 

6.75 

2.93 

41^ 

3.37 

3.22 

2.28 

142,400 

71.20 

85,440 

42.72 

7.12 

3.01 

4/4 

3.55 

3.30 

2.34 

150,000 

7500 

90,000 

45.00 

7.50 

3.10 

5 

3.75 

3.39 

2.40 

181  JOINTS  OR  CONNECTIONS  IN  IRON  CONSTRUCTION. 

JOINTS  OR  CONNECTIONS  IN  IRON  CONSTRUCTION. 
PROPORTIONS  OF  BOLTS,  NUTS,  RIVETS,  &c. 

Reft  re  nee. 

A  =  Sectional  area  of  bolt,  rivet,  or  pin. 
AI=  Sectional  area  of  all  rivets  in  a  joint. 
A%—  Sectional  area  of  one  plate. 
D  =  Diameter  of  bolt,  rivet,  or  pin. 

S=  Ultimate  resistance  to  shearing  of  material. 

T  =  Ultimate  resistance  to  tearing  of  material. 

TI=  Tensional  strain  on  joint,  &c. 

a  =  Number  of  times  that  a  bolt,  &c.,  will  have  to  be  sheared- 
(See  2  on  Fig.  258.) 

d  =  Distance  between  centres  of  rivets. 

k  =  Factor  of  safety. 

I  =  Overlap,  approximately  If  d  to  If  d. 
m  =  Number  of  rivets  in  a  joint. 

n  =  Number  of  lines  of  rivets  in  a  joint  at  right  angles  to  strain. 

t  =  Thickness  of  a  plate. 

RIVETS. 
Fig.  257. 

For  tension  in  direction  of  rivet: 


-J- 


T  0.7854 


For  shearing  at  right  angles  : 

If  at  one  place  D=    I -Tl  *__ 

N    S  0.7854 


If  at  two  places  D 


=         I  _  ?L 

>J   S  1.5 


__ 
.5708 

Approximately  :    I  =  3t        D  =  3t 


JOINTS  OR  CONNECTIONS  IN  IRON  CONSTRUCTION. 

PIN,  &o.,  IN  TIE  BARS. 

Fig.  258. 


PLATE  JOINTS. 

No.  I. — Plate  Joint  Overlapped,  four  lines  of  Rivets. 

Fig.  259. 

-£ «*.      ;  d  =--  D  +  -L  (0.7854  JD»n) 

o    ©.      d 

©  .#     ©i-         Approximately  c?  =  1.5i  to  2t 

i^-ct-©  cJ©--ci->i    ft  A    A 

©  (t)-^- 


4V 


^0.7854 


__ 

2mtS 


2. — Ptafe  Jbm^  Overlapped,  single  line  of  Rivet 
Fig.  260.     (Same  as  No.  1.) 


186 


JOINTS  OR  CONNECTIONS  IN  IRON  CONSTRUCTION. 


No.  3. — Plate  Joint  Overlapped,  two  lines  of  Rivets. 

Fig.  261.     (Same  as  No.  1.) 

o      o 


No.  4. — Fish  Joints,  two  lines  of  Rivets. 
Fig.  262. 


One  fish  plate.     (Same  as  No.  1.) 

Two  fish  plates. 

Thickness  of  each  fish  plate  =  J  t. 


/>_-!_  /__**. 

m   **    £1.570. 


L.5708 
(Otherwise  same  as  No.  1.) 


DIMENSIONS   OF   BOLTS   AND   NUTS. 


187 


DIMENSIONS  OF  BOLTS  AND  NUTS. 

(Whitworth's  proportions.) 
Figs.  263,  264,  265,  266,  267,  268,  269,  270,  and  271. 


Inch. 

3 

21 

2} 

21 
2 


If 


H 

i 

I 


A 

I 

A 


Dimension  of  Nuts  and  Heads. 


.  -^ 

Inch. 

p  ?» 

Inch. 

Inch. 

Inch. 

4J 

5.18 

5 

7.07 

4J 

4.76 

4J 

6.37 

3| 

4.33 

4J 

5.83 

3| 

3.89 

3| 

5.30 

3 

3.46 

3| 

4.76 

2f 

3.17 

3 

4.24 

2f 

3.03 

21- 

3.89 

^1 

2.88 

2| 

3.71 

2J 

2.59 

21 

3.53 

2 

2.30 

21 

3.18 

H 

2.16 

2 

2.82 

11 

1.87 

IF 

2.64 

1} 

1.73 

If 

2.29 

!& 

1.51 

H 

2.12 

1* 

1.38 

1A 

1.86 

1 

1.15 

1A 

1.67 

| 

1.01 

i 

1.41 

i 

0.86 

1 

1.23 

t 

0.86 

f 

1.06 

A 

0.64 

I 

1.06 

TV 

0.50 

A 

0.79 

f 

0.43 

A 

0.79 

Dia.  of  No.  Threads 
Core.       per  inch* 


Inch.    Inch, 


3 

2 

.57 

3. 

5 

1 

.50 

2f 

2 

.35 

'  3 

.5 

1 

.75 

2J 

2 

.13 

4, 

,0 

2 

.00 

21 

1 

.91 

4, 

,0 

2 

.12 

2 

1 

.69 

4, 

5 

2 

.25 

If 

1 

.58 

4 

.5 

2 

.37 

If 

1 

.47 

5, 

,0 

2 

.50 

If 

1 

.36 

5. 

;0 

2 

.75 

If 

1 

.25 

6 

0 

3 

.00 

If 

1 

.14 

6, 

,0 

5 

.25 

It 

1 

.08 

7, 

,0 

3 

.50 

H 

0 

.92 

7, 

,0 

4 

.00 

l 

0 

.81 

8. 

0 

5 

.00 

i 

0 

.70 

9. 

0 

6 

.00 

i 

0 

.59 

10 

,0 

6 

.00 

i 

0 

.48 

11, 

,0 

7 

.00 

9 
T6 

0 

.42 

11. 

0 

7 

.00 

J 

0 

.37 

12. 

0 

8 

.00 

TV 

0 

.31 

14, 

,0 

8 

.00 

1 

0 

.26 

16. 

0 

9 

.00 

* 

0 

.20 

18 

.0 

9 

.00 

i 

0 

.15 

20 

.0 

10 

.00 

188  STRAINS  IN  HORIZONTAL  AND  SLOPING  BEAMS. 

Fig.  272. 


Approximate  proportions  of  bolts,  nuts,  and 
beads  in  incbes: 

d  =  1.4  D  -\-  0.25  =  Inscribed  circle. 
h  =  D  =  Height  of  nut. 
/*!=  0.7  D  =  Height  of  head. 


COMPOUND  STRAINS  IN  HORIZONTAL  AND  SLOPING 
BEAMS. 

(Load  equally  distributed  or  between  supports.) 

Area  of  Cross-section  necessary  to  resist  a  Cross-breaking  and 
Compressive  Strain  in  Beams  acting  as  a  Boom  in  Trusses,  &c,, 
or  Beams  acting  as  Rafters,  &c. 

Reference. 

rii  =  Bending  moment  (See  Page  100.) 
C=  Compressive  strain.     (See  Roof  and  Simple  Trusses.) 
g  =  A  factor  depending  on  form  of  cross-section. 
/=  Moment  of  inertia  of  cross-section. 
8  =  Distance  from  neutral  axis  to  most  compressed  fibres. 
A  =  Sectional  area  of  beam,  &c. 
h  =  Depth  of  beam,  &c. 
p  =  Resistance  to  compression  with  safety  per  square  inch  of 

section. 

W=  Total  load. 
I  =  Length  of  beam,  &c. 


_ 


STRAINS  IN  HORIZONTAL  AND  SLOPING  BEAMS.  189 

For  horizontal  beams,  &c.  : 


For  sloping  beams,  &c.,  v  =  angle  between  horizontal  and  beam: 

W  r  1    /     1  \\  I  cos.  v~\ 

A  =  --  LT  (-sTnTV  +  sm-  v)  +  T2?T  J 


RAFTER  OF  A  ROOF  TRUSS. 
Fig.  273. 


EXAMPLE. 

Reference. 
W  =  2.5  tons.         0  =  2.8  tons.        I  =  10  feet.        v  =  26°  30' 

p  =.  5  tons  per  square  inch. 

We  will  assume  a  Phoenix  Go's  six-inch  beam  of  the  following 
dimensions:    h  =  6  inches;   A  =  4  inches;    J=  22.5 


showing  that  the  six-inch  beam  has  a  greater  sectional  area  than 
required. 

If  the  load  is  concentrated  at  the  apex  of  roof,  the  compressive 
strain  C=  2.8  tons,  and  the  area  necessary  to  resist  this  strain 

2  8 

would  be  (taking  p  at  five  tons  per  square  inch)  —  '-  —  —  0.56  sq. 

D 

inches,  provided  this  is  able  to  resist  buckling. 

By  comparing  this  with  the  above  result,  it  will  be  seen  how 
much  greater  the  sectional  area  will  have  to  be  to  resist  a  cross  - 
breaking  strain,  caused  by  the  load  being  distributed.  These 
remarks  also  apply  to  simple  trusses. 


190  STRAINS  IN  HORIZONTAL  AND  SLOPING  BEAMS. 

SIMPLE  TRUSS,  (BEAM  CONTINUOUS  OVER  STRUT.) 
Fig.  274. 


EXAMPLE. 

Reference. 
W  •=  20  tons.     I  =  20  feet,     v  =  15°    p  =  5  tons  per  sq.  inch. 

We  will  assume  a  Phoenix  Go's  twelve-inch  beam   of  the  fol 
lowing  dimensions: 

h  =  12  inches.  J=  275.92 

A  =  12.5  inches.  s  =  6  inches. 

275.92 

= 


0.5  X  12'  X  12.5 

m=0.0703xlXl202=  84.36    (See  Reaction  of  Supports.) 
C=  23.32  tons. 

84.36      \   ,  46.26 


0.306x12 


)  +23.32=-— —=9.25  inches. 


Consequently  the  sectional  area  of  the  twelve-inch  beam  is 
amply  sufficient. 

[NOTE.— The  formulas  for  horizontal  beams  are  also  applicable  to  rafters 
of  roof  trusses,  m  and  C  being  given.  For  the  bending  moments  (m) 
the  various  distances  are  the  horizontal  projections  of  those  on  the  rafter 
from  abutment  to  ridge. 

The  loregoing  formulas  also  apply  to  beams  under  a  cross-creaking  and 
tensional  strain.  If  the  truss  (Fig.  274)  is  inverted,  the  horizontal  member 
will  be  in  tension.  Hence,  insert  the  resistance  of  the  material  to  tension 
instead  of  compression,  and  put  tensional  for  compressive  strain;  other 
wise,  the  formulas  remain  the  same.] 


WEIGHT  OF  MOVING   LOADS. 


191 


WEIGHT   OF   MOVING    LOADS. 

Variable  and  Accidental  Loads. 
(Weight  of  construction  not  included.) 


Character  of 
structure. 

How  loaded. 

Weight  in  Ibs.  per  square 
foot  of  surface. 

Street  bridges  for 
horse  cars  and 
heavy  traffic. 

Crow'd  with  per 
sons. 

Minimum  

40  Ibs. 
120    " 

80    " 

Maximum   

Average 

Street  bridges  for 
general  traffic, 
foot  passengers, 
&c. 

Persons,  animals, 
and  wagons. 

Public   travel.... 
Private  travel... 
Heavy  business 
wagons 

80  Ibs. 

40    " 

80    " 
40    " 

Light     business 
wagons  

Floors,  &c  • 

Crowded  public 
places. 

Dwellings  

Minimum 

40  Ibs. 

120    " 
80    " 
40    " 

80    " 
100    " 

200    " 
250    " 
f200 
\    to     " 
(400 
80    " 

Maximum  
Average 

Churches,    court 
rooms,  theatres, 
and  ball-rooms. 
Storage  of  grain... 
General  merchan 
dise  

Warehouses  

Factories.  

Hay-  lofts  

192 


STATIC  AND  MOVING  LOADS  ON  BRIDGES. 


STATIC  AND  MOVING   LOADS   ON    BRIDGES   OF 
WROUGHT  IRON. 

The  following  table  gives  an  approximate  weight  per  lineal 
foot  in  pounds  of  the  static  load  or  weight  of  construction  complete 
for  Single-Line  Railway  Bridges,  supported  at  the  ends,  from  ten 
to  four  hundred  feet  span;  also  the  weight  of  the  moving  load 
per  lineal  foot  of  span,  based  on  the  assumption  that  the  heaviest 
locomotives  exert  a  pressure  of  three  thousand  pounds  per  lineal 
foot  between  their  extreme  bearings. 

The  table  is  applicable  in  computing  the  strains  in  all  trusses 
with  parallel  booms  mentioned  in  this  work. 

Weight  of  Construction  and  Moving  Load  of  Wrought- Iron  Single- 
Line  Railway  Bridges  for  the  heaviest  traffic. 

(From  20  to  400  feet  span.) 


Weight  of  construction  complete, 
including  cross-ties  and  rails. 

Weight  of  moving  load  equal  to  3,000 
Ibs.  per  lineal  foot  of  load. 

£ 

Weight  in 

d 

Weight  in 

d 

Weight  in 

jj 

Weight 
in  Ibs. 

.2 

Ibs.  per 

.2 

Ibs.  per 

.2 

Ibs.  per 

.2 

a 

lineal  foot 

a 

lineal  foot 

a 

lineal  foot 

23 

per  lin. 

1 

of  span. 

a 

rSl 

of  span. 

c3 
A 
V) 

of  span. 

a 

CO 

span. 

10 

427 

210 

1,891 

10 

6,300 

210 

2,535 

20 

500 

220 

1,964 

20 

5,370 

220 

2,495 

30 

573 

230 

2,037 

30 

4,250 

230 

2,455 

40 

646 

240 

2,110 

40 

3,780 

240 

2,375 

50 

719 

250 

2,183 

50 

3,550 

250 

L',335 

60 

792 

260 

2,256 

60 

3,400 

260 

2,290 

70 

865 

270 

2,329 

70 

3,300 

270 

2,245 

80 

938 

280 

2,402 

80 

3,250 

280 

2,200 

90 

1,011 

290 

2,475 

90 

3,180 

290 

2,160 

100 

1,084 

300 

2,548 

100 

3,120 

300 

2,120 

110 

1,157 

310 

2.621 

110 

3,050 

310 

2,080 

120 

1,230 

320 

2,694 

120 

3,000 

320 

2,045 

130 

1,303 

330 

2,767 

130 

2,930 

330 

2,010 

140 

1,380 

340 

2,840 

140 

2,880 

340 

1,975 

150 

1,453 

350 

2,913 

150 

2,820 

350 

1,940 

160 

1,526 

360 

2,986 

160 

2,760 

360 

1,910 

170 

1,599 

370 

3,059 

170 

2,700 

370 

1,880 

180 

1,672 

380 

3,132 

180 

2,655 

380 

1,850 

190 

1,745 

390 

3,205 

190 

2,615 

390 

1,820 

200 

1,818 

400 

3,278 

200 

2,575 

400 

1,890 

STATIC  AND  MOVING  LOADS  ON  BRIDGES. 


193 


The  following  gives  the  actual  weight  of  some  well-known 
Bridges  (single  line)  in  America,  Germany,  and  England: 


Name  of  Bridge. 

System. 

a 

Weight  of  con 
struction  per 
lineal  foot. 

6  <jj 

<4_T3~0 

^ 

.5  *•"£ 

c«  O  ^ 
CO 

a 

CO 

Lbs. 

Lbs. 

Lbs. 

"Brenz,"  near 
Konigsbronn... 

"Colomak"  

"Iser,"  near  Mu 
nich  

|      Open  Web,     ^ 
{  parallel  booms,  f 

63.0 
111.0 
164.7 

760 
1,090 
1,770 

3,131 
3,067 
3656 

7,530 
9,516 
8,532 

"Donau,"  near 
Ingolstadt  

"Elb,"nearMei- 

ssen 

'< 

178.0 
179  0 

1,954 
1  324 

3,312 
2  783 

8,532 
10390 

"Rhine,"  near 
Mainz 

{"Pauli's,"  par-  ") 
abolic  arched    > 

345  0 

2  170 

1  970 

11  660 

"Royal  Albert," 
near  Saltash... 

"Boyne"  

booms.         J 
Lattice..**...  

455.0 
264.0 

4,418 
3225 

2,240 

9,954 

"  Leven  "  

36  0 

566 

"Kent"  

M 

36  0 

580 

"Harper's  Ferry" 

Truss  

124.0 

770 

13 


MISCELLANEOUS. 


(195) 


QEOMETRY. 


LONGIMETEY   AND   PLANIMETRY. 
(Lines  and  Areas.) 

Reference. 
A  =  Area. 

-  =  Periphery  of  circle  =  3.14159  when  diameter  =  1. 
r  =  Radius  of  circle. 
G  =  Length  of  cord  of  segment. 
p  =  Circumference  of  circle  for  given  diameter. 
I  =  Length  of  circle  arc,  &c. 
h  =  Height  of  segment. 

v  =  Angles,  expressed  in  decimals,  as  15°  30/=  15.5. 
For  other  designations,  see  Figuies. 

[NOTE.— Always  use  the  same  unit  for  dimensions.] 


•t 


Values  cj  TT. 

-  =  \14159  x 

2~  =  6.28319  ~  =  1.04720 

=  0.31831  TT 

*•  —  =    0.78540 

=  0.15915  ;r 

--=    0.52360 


1 


__—  =  0.10132  -2=    9.86960 

-3  _  31.00628 

—  =0.63662  ^1=    L77245 

&-=    1.46459 


(197) 


198 


LONGIMETRY. 


Fig.  275. 


p  = 


Fig.  276. 


360°   1  "  "    360° 
I 


180° 


v  =  -  180° 
TTr 

180°       ^ 


7T 


r.  277. 


'  =  2(180°  — 


Fig.  278. 


8h          "    2h 


.  279. 


LONGIMETRY. 


199 


Fig.  280. 


Fig.  281. 
Ellipse. 


Fig.  282. 


[7)2 
1+^ 

-  +  1 

256  n    '"J 

When  n  = , — ; — 

a  -\-  b 


b  =  \/az—  c* 
a  =  \/62-f  c2 


Fig.  283. 


.  284. 


c2— a2— Z>2 
~26~ 


200 


PLANIMETRY. 


85.     (Circle  plane.) 


286.    (Circle  ring.) 


.  287.     (Sector.)  li 


=  0.008727  w2. 

/  360°     . 
r==N/~T~~~ 


288.     (Segment.) 


A  = 


?;  —  sin.v)  — 


(0.017453  v  —  sin.  »)  — - 

2 


Fig. 28$.  (Circle  ring  sector) 


360°    V1 
:  0.008727  ^(ry2  — r22) 


PLANIMETRY. 


Fig.  290.     (Ellipse.) 


'  A  ==  nab 


Fig.  291.     (Square.) 


Fig.  292.     (Rectangle.) 


A  =  a2 


Fig. 293.  (Parallelogram.) 


=  a    sn.  v 


294.     (Triangle.) 


A  =  — —  = be  sin.  v 

2  2 

c2  sin.  v  sin.  Vj 


2  sin.  v2 

When  the  three  sides  are  given: 
Let  a  +  b  +  c  =  s 


CENTER   OP   GRAVITY   OF   PLANES. 


CENTER  OF  GRAVITY  OF  PLANES. 
Reference. 

x  =  Distance  from  a  fixed  base  to  center  of  gravity. 
r  =  Radius. 
c  =  Chord. 
b,p,  h  =  Dimensions. 
A  =  Area. 
v  =  Angle. 


Fig.  295.     (Quadrangle.) 


Fig.  296.     (Triangle.) 


Fig.  297.    (Half  circle,  or 
elliptic  plane.) 


a  and  b  parallel. " 

h          h    (  b  —  a  \ 
X         2 6"  Vfi+o  J 


— -  =  radius  =  r 

£l 

x  =  0.4244r 


Fig.  298.  (Concentric  ring.) 


4     sin.  Ji;      r3  —  r^ 

""  ~3         v          r2  — r,2 


CENTER  OF  GRAVITY  OF  PLANES. 


203 


Fig.  299.  (Circle,  or  elliptic 
arc.) 


re      2  sin. 


V 


JF%,300.  (Half  circumfer 
ence  of  circle  or  ellipse.) 


x  = r  =  0.6366r 

7T 


Fig.  301.     (Circle  sector. 


4      sin.  -Jv 

* 


Fig.  302.  (Circle  segment.) 


.  =  Area. 


Fig.  303.     (Parabola.) 


204 


CENTER  OF  GRAVITY  OF  PLANES. 


Fig.  305.  (Half  parabola.) 


Fig.  305. 


Of  any  section,  composed  of  any 
number  of  simple  figures: 

Additional  Reference. 

A,  -4,,  -4//=  Sectional  areaof  simple 

figures. 

X  =  Distance  from  center  of 
gravity  of  whole  sec 
tion  to  axis  ran. 

xt  x/t  £//  =  Distance  from  center  of 
gravity  of  a  simple 
figure  to  a  fixed  axis 
mn. 

Y  —  ^X  ~^~  ^/x/~^~  A//x/  +  &c- 


TRIGONOMETRICAL  FORMULAS. 


205 


TRIGONOMETRICAL  FORMULAS. 

Reference. 

a,  6,  c  =  Length  of  sides. 
A,  Bt  C=  Angles  opposite  to  a,  &,  c  respectively. 

Right  Angle  Triangle. 

Fig.  306. 


cos.  C 
b  =  a  cos.  C 
b  =  c  cot.  (7 
b  =  a  sin.  B 
b  =  c  tang.  B 
c  =  b  tang.  0 
c  =  a  sin.  0 


Tang.  (7=4-== n 

b         cos.  0 


"cot.  (7 


Cotang.  C= 
Secant  (7  = 
Cosec.  C  = 


cos.  (7 
sin.  C 

1 

cos.  (7 

1 

sin.  0 


~  tang.  C 


Cos.(7=— - 
a 


Oblique  Angle  Triangle. 

Fig.  307. 


B) 


a  =  \/62  +  c2  —  26c  cos. 
c  sin.  5 


Sin.  (7= 
Sin.  4  = 


c  sin. 


6 
a  sin.  C 


b 

c  sin.  A 
a 


TRIGONOMETRICAL    FUNCTIONS. 
NATURAL  SINE 


Minutes. 

Deg. 

0 

5 

10 

15 

20 

25 

30 

0 

.00000 

.00145 

.00291 

.00436 

.00582 

.00727 

.00873 

1 

.01745 

.01891 

.02036 

.02181 

.02327 

.02172 

.02618 

2 

.03490 

.03635 

.03781 

.03926 

.04071 

.04217 

.04302 

3 

.05234 

.05379 

.05524 

.05669 

.05814 

.05960 

.06103 

4 

.00976 

.07121 

.07266 

.07411 

.07556 

.07701 

.07846 

5 

.08716 

.08860 

.09005 

.09150 

.09295 

.09440 

.09585 

6 

.10453 

.10597 

.10742 

.10887 

.11031 

.11176 

.11320 

7 

.12187 

.12331 

.12476 

.12620 

.12764 

.12908 

.13053 

8 

.13917 

.14061 

.14205 

.14349 

.14493 

.14637 

.14781 

9 

.15643 

.15787 

.15931 

.16074 

.16218 

.16361 

.16505 

10 

.17365 

.17508 

.17651 

.17794 

.17937 

.18081 

.18224 

11 

.19081 

.19224 

.19366 

.19509 

.19652 

.19794 

.19937 

12 

.20791 

.20933 

.21076 

.21218 

.21360 

.21502 

.21644 

13 

.22495 

.22(537 

.22778 

.22')20 

.23062 

.23203 

.23345 

14 

.24192 

.24333 

.24474 

.24615 

.21756 

.24897 

.25038 

15 

.25882 

.20022 

.25163 

.26303 

.20443 

.20584 

.26724 

16 

.27564 

.27704 

.27843 

.27983 

.28123 

.28202 

.28402 

17 

59237 

.29376 

.29515 

.29654 

.29793 

.29932 

.30071 

18 

.30902 

.31040 

.31178 

.31316 

.31454 

.31593 

.31730 

19 

.32567 

.32694 

.32832 

.32969 

.33106 

.33244 

.33381 

20 

,342i)2 

.34339 

.34475 

.34612 

.34748 

.34884 

.35021 

21 

.35837 

.35973 

.36108 

.36244 

.36379 

.36515 

.36650 

22 

.37461 

.37595 

.37730 

.37865 

.37999 

.38134 

.38268 

23 

.39073 

.39207 

.39341 

.39474 

.39608 

.39741 

.39875 

24 

.40674 

.40806 

.40939 

.41072 

.412  4 

.41337 

.41469 

25 

.42232 

.42394 

.42525 

.42657 

.42788 

.42920 

.43051 

26 

.43837 

.43968 

.44098 

.44229 

.44359 

.44494 

.44620 

27 

.45399 

.45529 

.45658 

.45787 

.45917 

.46046 

.46175 

28 

.46947 

.47076 

.47204 

.47332 

.47460 

.47588 

.47716 

29 

.48481 

.48608 

.48735 

.48862 

.48989 

.49116 

.49242 

30 

.50000 

.50126 

.50252 

.50377 

.50503 

.50628 

.50754 

31 

.51504 

.51628 

.51753 

.51877 

.52002 

.52120 

.52250 

32 

.52992 

.53115 

.53238 

.53361 

.53484 

.53607 

.53730 

33 

.54464 

.54586 

.54708 

.54829 

.54951 

.55072 

.55194 

34 

.55919 

.56040 

.56160 

.56280 

.56401 

.56521 

.56641 

35 

.57358 

.57477 

.57596 

.57715 

.57833 

.57952 

.58070 

36 

.58779 

.58869 

.59014 

.59131 

.59248 

.59365 

.59482 

37 

.00182 

.60298 

.60414 

.60529 

.60(545 

.60761 

.60876 

38 

.61566 

.61681 

.61795 

.61909 

.62024 

.62138 

.62251 

39 

.62932 

.63045 

.63158 

.63271 

.63383 

.63496 

.63608 

40 

.64279 

.64390 

.64501 

.64612 

.64723 

.64834 

.64945 

41 

.65606 

.65716 

.65825 

.65935 

.66044 

.66153 

.66202 

42 

.66913 

.67221 

.67129 

.67237 

.67344 

.67452 

.'67559 

43 

.68200 

.68306 

.68412 

.68518 

.68624 

.68730 

.68835 

44 

.69466 

.69570 

.69675 

.69779 

.69883 

.69987 

.70091 

Deg. 

60 

55 

50 

45 

40 

35 

30 

Minutes. 

NATURAL  COSINE. 


TRIGONOMETRICAL   FUNCTIONS. 
NATURAL  SINE. 


Minutes. 


Deg. 

35 

40 

45 

50 

55 

GO 

.01018 

.01164 

.01309 

.01454 

.01000 

.01745 

89 

.02703 

.02908 

.03054 

.03199 

.03345 

.03490 

88 

.04507 

.04053 

.04798 

.04913 

.05088 

.05234 

87 

.06250 

.00395 

.06540 

.06685 

.06831 

.00970 

86 

.07991 

.08130 

.08281 

.08426 

.08571 

.08710 

85 

.09729 

.09874 

.10019 

.10104 

.10308 

.10453 

84 

.11405 

.11009 

.11754 

.11898 

.12043 

.12187 

83 

.13197 

.13341 

.13485 

.13029 

.13802 

.13917 

82 

.14025 

.15009 

.15212 

.15356 

.15500 

.15043 

81 

.10048 

.10792 

.16935 

.17078 

.17222 

.17305 

80 

.18307 

.18509 

.18652 

.18795 

.18938 

.19081 

79 

.20079 

.20222 

.20364 

.20507 

.20649 

.20791 

78 

.21780 

.21928 

.22070 

.22212 

.22353 

.22495 

77 

.23480 

.23627 

.23769 

.23910 

.24051 

.24192 

76 

.25179 

.25320 

.2.5460 

.25601 

.25741 

.25882 

75 

.23804 

.27004 

.27144 

.27284 

.27421 

.27504 

74 

.28541 

.28080 

.28820 

.28959 

.29098 

.29237 

73 

.30209 

.30348 

.30486 

.30625 

.30703 

.30902 

72 

.31808 

.32006 

.32144 

.32282 

.32419 

.32057 

71 

.33518 

.33655 

.33792 

.33929 

.34005 

.34202 

70 

.35157 

.35293 

.35429 

.35565 

.35701 

.35837 

69 

.30785 

.36921 

.37056 

.37191 

.37320 

.37401 

68 

.38403 

.38537 

.38671 

.38805 

.38939 

.39073 

67 

.40008 

.40141 

.40275 

.40408 

.40541 

.40074 

66 

.41002 

.41734 

.41866 

.41998 

.42130 

.422i2 

65 

.43182 

.43313 

.43445 

.43575 

.43700 

.43837 

64 

.44750 

.44880 

.45010 

.45140 

.45209 

.45399 

03 

.40304 

.46433 

.46561 

.46690 

.40819 

.40947 

62 

.47844 

.47971 

.48099 

.48226 

.48354 

.48481 

61 

.49309 

.49495 

.49622 

.49748 

.49874 

.50000 

60 

.50879 

.51004 

.51129 

.51254 

.51379 

.51504 

59 

.52374 

.52498 

.52821 

.52745 

.52809 

.52992 

58 

.53853 

.53975 

.54097 

.54220 

.54342 

.54404 

57 

.55315 

.55436 

.55557 

.55678 

.55799 

.55919 

56 

.50700 

.56880 

.57000 

.57119 

.572:38 

.57358 

55 

,58189 

.58307 

.58425 

.58543 

.58001 

.58779 

54 

.59599 

.59716 

.59832 

.59949 

.60065 

.00182 

53 

.60991 

.61107 

.61222 

.61337 

.61451 

.61560 

52 

.62305 

.62479 

.62.395 

.62706 

.62819 

.62932 

51 

.63720 

.63832 

.63944 

.64056 

.64107 

.64279 

50 

.65055 

.65166 

.65276 

.65386 

.65496 

.65000 

49 

.66371 

.66480 

.66588 

.66097 

.66805 

.00913 

48 

.67600 

.67773 

.67880 

.07987 

.68093 

.08200 

47 

.68941 

.69046 

.69151 

.09256 

.69361 

.69466 

46 

.70195 

.70238 

.70401 

.70505 

.70008 

.70711 

45 

23 

21 

15 

10 

5 

0 

MintitPs. 


NATURAL  COMNE. 


TRIGONOMETRICAL    FUNCTIONS. 
NATURAL  SINE. 


Minutes. 

Deg. 

0 

5 

10 

15 

20 

25 

30 

1 

45 

.70711 

.70813 

.70916 

.71019 

.71121 

.71223 

.71325 

46 

.71934 

.72035 

.72136 

.72230 

.72337 

.72437 

.72,337 

47 

.73135 

.73234 

.73333 

.73432 

.73531 

.73629 

.73728 

48 

.74314 

.74412 

.74509 

.74000 

.74703 

.74799 

.74890 

49 

.75471 

.75506 

.75661 

.75756 

.75851 

.75940 

.70041 

50 

.76604 

.76698 

.76791 

.76884 

.76977 

.77070 

.77102 

51 

.77715 

.77806 

.77897 

.77988 

.78079 

.78170 

.78201 

52 

.78801 

.78891 

.78980 

.79069 

.79158 

.79247 

.79335 

53 

.79804 

.79951 

.80038 

.80125 

.80212 

.80299 

.80380 

54 

.80902 

.80987 

.81072 

.81157 

.81212 

.81327 

.81412 

55 

.81915 

.81999 

.82082 

.82105 

.822i8 

.82330 

.82413 

56 

.82904 

.82985 

.83006 

.83147 

.83228 

.83308 

.83389 

57 

.83807 

.83946 

.84023 

.84104 

.84182 

.84201 

.84339 

58 

.84805 

.84882 

.84959 

.85035 

.85112 

.85188 

.85204 

59 

.85717 

.85792 

.85866 

.85941 

.86015 

.80089 

.80103 

60 

.8(5003 

.80075 

.80748 

.80820 

.86892 

.80904 

.87036 

61 

.87402 

.87532 

.87003 

.87073 

.87743 

.87812 

.87882 

62 

.88295 

.88363 

.88431 

.88499 

.88566 

.88634 

.88701 

63 

.89101 

.89167 

.89232 

.89238 

.89303 

.89428 

.89493 

64 

.89879 

.89943 

.90007 

.90070 

.90133 

.90190 

.90259 

65 

.90631 

.90692 

.90753 

.90814 

.90875 

.90936 

.90996 

66 

.91355 

.91414 

.91472 

.91531 

.91590 

.91648 

.91706 

67 

.92.)50 

.92107 

.92164 

.92220 

.92276 

.92332 

.92388 

68 

.92718 

.92773 

.92827 

.92381 

.92935 

.92088 

.93042 

69 

.93358 

.93410 

.93462 

.93514 

.93565 

.93016 

.93667 

70 

.93969 

.94019 

.94068 

.94118 

.94167 

.94215 

.94264 

71 

.94552 

.94599 

.94046 

.94093 

.94740 

.94786 

.94832 

72 

.95106 

.95150 

.95191 

.95240 

.95284 

.95328 

.95372 

73 

.95630 

.95673 

.95715 

.95757 

.95799 

.95841 

.95882 

74 

.96196 

.96166 

.90206 

.90246 

.90235 

.90324 

.96363 

75 

.96593 

.96630 

.90667 

.96705 

.90742 

.90778 

.96815 

76 

.97030 

.97065 

.97100 

.97134 

.97109 

.97203 

.97237 

77 

.97437 

.97470 

.97502 

.97534 

.97566 

.97598 

.97030 

78 

.97815 

.97845 

.97875 

.97905 

.97934 

.97963 

.97992 

79 

.98163 

.98190 

.98218 

.98245 

.98272 

.98299 

.98325 

80 

.98481 

.98506 

.98531 

.98506 

.98580 

.98004 

.98629 

81 

.98769 

.98791 

.98814 

.98836 

.98858 

.98880 

.98902 

82 

.99027 

.99047 

.99067 

.99087 

.99106 

.99125 

.99144 

83 

.99235 

.99272 

.99290 

.99307 

.99324 

.99341 

.99357 

84 

.99452 

.99467 

.99482 

.99497 

.99511 

.99526 

.99540 

85 

.99619 

.99632 

.99644 

.99657 

.99668 

.99080 

.99092 

86 

.99756 

.99766 

.99776 

.99786 

.99795 

.99804 

.99813 

87 

.99863 

.99870 

.99878 

.99885 

.99892 

.99898 

.99905 

88 

.99939 

.99944 

.99949 

.99953 

.99958 

.99962 

.99906 

89 

.99985 

.99987 

.99989 

.99991 

.99993 

.99995 

.99996 

60 

55 

50 

45 

40 

35 

30 

Deg 

Minutes. 

NATURAL  COSINE. 


TRIGONOMETRICAL    FUNCTIONS. 
NATURAL  SINE. 


Minutes. 

35 

40 

45 

50 

55 

60 

Deg. 

.71427 

.71529 

.71630 

.71732 

.71833 

.71934 

44 

.72637 

.72737 

.72837 

.72937 

.73036 

.73135 

43 

.73826 

.73924 

.74022 

.74123 

.74217 

.74314 

42 

.74992 

.75088 

.75184 

.75280 

.75375 

.75471 

41 

.76135 

.76229 

.76323 

.76417 

.76511 

.76004 

40 

.77255 

.77347 

.77439 

.77531 

.77023 

.77715 

39 

.78351 

.78442 

.78532 

.78622 

.78711 

.78801 

38 

.79424 

.79512 

.79300 

.79088 

.79776 

.79804 

37 

.80472 

.80558 

.89644 

.80730 

.80816 

.80902 

36 

.81496 

.81580 

.81664 

.81748 

.81832 

.81915 

35 

.82495 

.82577 

.82659 

.82741 

.82822 

.82904 

34 

.83469 

.83549 

.83629 

.83708 

.83788 

.83807 

33 

.84417 

.84495 

.84573 

.84050 

.84728 

•84805 

32 

.85340 

.85416 

.85491 

.85507 

.85642 

.85717 

31 

.86237 

.86317 

.80384 

.80457 

.86530 

.80003 

30 

.87107 

,87178 

.87250 

.87321 

,87391 

.87462 

29 

.87959 

,88020 

.88089 

.88158 

.88226 

.88295 

28 

.88768 

.88835 

•88902 

.88968 

.89035 

.89101 

27 

.89558 

.8962:3 

.89687 

.89752 

.89816 

.89879 

26 

.90321 

.90383 

.90446 

.90507 

.90569 

.90631 

25 

.91056 

.91116 

.91176 

.91236 

.91295 

.91355 

24 

.91764 

.91822 

.91879 

.91936 

.91994 

.92050 

23 

.92444 

.92499 

.92554 

.92609 

.92064 

.92718 

22 

.93095 

.93148 

.93201 

.93253 

.93306 

.93358 

21 

.93718 

.93709 

.93819 

.93809 

.93919 

.93969 

20 

.94313 

.94301 

.94409 

.94457 

.94504 

.94552 

19 

.94878 

.94924 

.94970 

.95015 

.95001 

.95106 

.95415 

.95459 

.95502 

.95545 

.95588 

.95030 

17 

.95923 

.95904 

.96005 

.9GC46 

.90086 

.90120 

16 

.96402 

.96440 

.96479 

.96517 

.90555 

.90593 

15 

.96851 

.90887 

.96923 

.96959 

.90994 

.97030 

14 

.97271 

.97304 

.97338 

.97371 

.97404 

.97437 

13, 

.97661 

.97092 

.97723 

.97754 

.97784 

.97815 

12 

.98021 

.98050 

.98079 

.98107 

.98135 

.98103 

11 

.98352 

.98378 

.98404 

.98430 

.98455 

.98481 

10 

.98652 

.98676 

.98700 

.98723 

.98746 

.98709 

9 

.98923 

.98944 

.98965 

.98986 

.99006 

.99027 

8 

.99163 

.99182 

.99200 

.99219 

.99237 

.99255 

7 

.99374 

.90390 

.99406 

.99421 

.99437 

.99452 

6 

.99553 

.99567 

.99580 

.99594 

.99007 

.99019 

5 

.99703 

.99714 

.99725 

.99736 

.99746 

.99756 

4 

.99822 

.99831 

.99839 

.99847 

.99855 

.99863 

3 

.99911 

.99917 

.99923 

.99929 

.99934 

.99939 

2 

.99969 

.99973 

.99976 

.99979 

.99982 

.99985 

1 

.99997 

.99998 

.99999 

1.00000 

1.00000 

1.00000 

0 

25* 

20 

15 

10 

5 

0 

Minutes. 

NATURAL  COSINE. 


210 


TRIGONOMETRICAL    FUNCTIONS. 
NATURAL  TANGENT. 


Minutes. 

Deg. 

0 

5 

10 

15 

20 

25 

30 

0 

0.0000 

0.0014 

0.0029 

0.0044 

0.0058 

0.0073 

0.0087 

1 

0.0175 

0.0189 

0.0204 

0.0218 

0.0233 

0.0247 

0.0262 

2 

0.0349 

0.0364 

0.0378 

0.0393 

0.0407 

0.0422 

0.0437 

3 

0.0524 

0.0539 

0.0553 

0.0508 

0.0582 

0.0597 

0.0612 

4 

O.OG99 

00714 

0.0728 

0.0743 

0.0758 

0.0772 

0.0787 

5 

0.0875 

0.0889 

0.0904 

0.0919 

0.0933 

0.0948 

0.0963 

6 

0.1051 

0.1066 

0.1080 

0.1095 

0.1110 

0.1125 

0.1139 

7 

0.1228 

0.1243 

0.1257 

0.1272 

0.1287 

0.1302 

0.1316 

8 

0.1405 

0.1420 

0.1435 

0.1450 

0.1465 

0.1480 

0.1495 

9 

0.1584 

0.1599 

0.1014 

0.1029 

0.1644 

0.1058 

0.1073 

10 

0.1763 

0.1778 

0.1793 

0.1808 

0.1823 

0.1838 

0.1853 

11 

0.1944 

0.1959 

0.1974 

0.1989 

0.2004 

0.2019 

0.2034 

12 

0.2120 

0.2141 

0.2150 

0.2171 

0.2186 

0.2202 

0.2217 

13 

((.2309 

0.2324 

0.2339 

0.2355 

0.2370 

0.2385 

0.2401 

14 

0.2493 

0.2509 

0.2524 

0.2540 

0.2555 

0.2571 

0.2586 

15 

0.2679 

0.2695 

0.2711 

0.2726 

0.2742 

0.2758 

0.2773 

16 

0.2867 

0.2883 

0.2899 

0.9915 

0.2930 

0.2946 

0.2962 

17 

0.3057 

0.3073 

0.3089 

0.3105 

0.3121 

0.3137 

0.3153 

18 

0.3249 

0.3265 

0.3281 

0.3297 

0.3314 

0.3330 

0.3346 

10 

0.3443 

0.34(50 

0.3470 

0.3492 

0.3508 

0.3525 

0.3541 

20 

0.3640 

0.3656 

0.3073 

0.3089 

0.3706 

0.3722 

0.3739 

21 

0.3839 

0  3855 

0.3872 

0.3889 

0.3905 

0.3922 

0.3939 

22 

0.4040 

0.4057 

0.4074 

0.4091 

0.4108 

0.4125 

0.4142 

23 

0.4245 

0.4262 

0.4279 

0.4296 

0.4314 

0.4331 

0.4348 

24 

0.4452 

0.4470 

0.4487 

0.4505 

0.4522 

0.4540 

0.4557 

25 

0.4663 

0.4681 

0.4698 

0.4716 

0.4734 

0.4752 

0.4770 

26 

0.4877 

0.4895 

0.4913 

0.4931 

0.4950 

0.4968 

0.4986 

27 

0.5095 

0.5114 

0.5132 

0.5150 

0.5169 

0.5187 

0.5206 

28 

0.5317 

0.5336 

0.5354 

0.5373 

0  5392 

0.5411 

0.5430 

29 

0.5543 

0.5502 

0.5581 

0.5600 

05619 

0.5638 

0.5658 

30 

0.5774 

0.5793 

0.5812 

0.5832 

0.5851 

0.5871 

0.5891 

31 

0.6008 

0.6028 

0.0048 

0.6068 

0.6088 

0.0108 

0.6128 

32 

0.6249 

0.6269 

0.0289 

0.6309 

0.0330 

0.0350 

0.6371 

33 

0.64!)4 

0.6515 

0.0535 

0.0550 

0.6577 

0.6598 

0.6619 

34 

0.0745 

O.G70G 

0.6787 

0.6809 

0.6830 

*  0.0851 

0.0873 

35 

0.7002 

0.7024 

0.7045 

0.7007 

0.7089 

0.7111 

0.7133 

36 

0.7205 

0.7288 

0.7310 

0.7332 

0.7355 

0.7377 

0.7400 

37 

0.7530 

0.7558 

0.7581 

0.7604 

0.7027 

0.7050 

0.7073 

38 

0.7813 

0.7836 

0.78GO 

0.7883 

0.7907 

0.7931 

0.7954 

39 

0.8098 

0.8122 

0.8146 

0.8170 

0.8195 

0.8219 

0.8243 

40 

0.8391 

0.8410 

0.8441 

0.8466 

0.8491 

0.8510 

0.8541 

41 

0.8693 

0.8718 

0.8744 

0.8770 

0.8795 

0.8821 

0.8847 

42 

0.9004 

0.9030 

0.9057 

0.9083 

0.9110 

0.9137 

0.9103 

43 

0.9325 

0.9352 

0.9380 

0.9407 

0.9434 

0.9402 

0.9490 

44 

0.9057 

0.9085 

0.9713 

0.9742 

0.9770 

0.9798 

0.9827 

60 

55 

50 

45 

40 

35 

30 

Deg. 

Minutes. 

NATCBAL  COTANGENT. 


TRIGONOMETRICAL   FUNCTIONS. 
NATURAL  TANGENT. 


Minutes. 


Deg. 

35 

40 

45 

50 

55 

60 

0.0102 

0.0116 

0.0131 

0.0145 

0.0160 

0.0175 

89 

0.0276 

0.0291 

0.0305 

0.0320 

0.0335 

0.0349 

88 

0.0451 

0.0466 

0.0480 

0.0495 

0.0509 

0.0524 

87 

0.0626  . 

0.0641 

0.0655 

0.0670 

0.0685 

0.0699 

86 

0.0802 

0.0816 

0.0831 

0.0846 

0.0860 

0-0875 

85 

0.0978 

0.0992 

0.1007 

0.1022 

0.1036 

0.1051 

84 

0.1154 

0.1169 

0.1184 

0.1198 

0.1213 

0.1228 

83 

0.1331 

0.1346 

0.1361  * 

0.1376 

0.1391 

0.1405 

82 

0.1509 

0.1524 

0.1539 

0.1554 

0.1569 

0.1584 

81 

0.1688 

0.1703 

0.1718 

0.1733 

0.1748 

0.1763 

80 

0.1868 

0.1883 

0.1899 

0.1914 

0.1929 

0.1944 

79 

0.2050 

0.2065 

0.2080 

0.2095 

0.2110 

0.2126 

78 

0.2232 

02247 

0.2263 

0.2278 

0.2293 

0.2309 

77 

0.2416 

0.2432 

0.2447 

0.2462 

0.2478 

0.2493 

76 

0.2602 

0.2617 

0.2633 

0.2648 

0.2664 

0.2679 

75 

0.2789 

0.2805 

0.2820 

0.2836 

0.2852 

0.2867 

74 

0.2978 

0.2994 

0.3010 

0.3026 

0.3041 

0.3057 

73 

0.3169 

0.3185 

0.3201 

0.3217 

0.3233 

0.3249 

72 

0.3362 

0.3378  " 

0.3394 

0.3411 

0.3427 

0.3443 

71 

0.3558 

0.3574 

0.3590 

0.3607 

0.3623 

0.3640 

70 

0.3755 

0.3772 

0.3789 

0.8805 

0.3822 

0.3839 

69 

0.3956 

0.3973 

0.3990 

0.4006 

0.4023 

0.4040 

68 

0.4159 

0.4176 

0.4193 

0.4210 

0.4228 

0.4245 

67 

0.4365 

0.4383 

0.4400 

0.4417 

0.4435 

0.4452 

66 

0.4575 

0.4592 

0.4010 

0.4628 

0.4645 

0.4(363 

65 

0.4788 

0.4805 

0.4823 

0.4841 

0.4859 

0.4877 

64 

0.5004 

0.5022 

0.5040 

0.5059 

0.5077 

0.5095 

63 

0.5224 

0.5243 

0.5261 

0.5280 

0.5298 

0.5317 

62 

0.5448 

0.5467 

0.5486 

0.5505 

0.5524 

0.5543 

61 

0.5677 

0.5696 

0.5715 

0.5735 

0.5754 

0.5774 

60 

0.5910 

0.5930 

0.5949 

0.5969 

0.5989 

0.6008 

59 

0.6148 

0.6168 

0.6188 

0.6208 

0.6228 

0.6249 

58 

0.6391 

0.6412 

0.6432 

0.6453 

0.6473 

0.6494 

57 

0.6640 

0.6661 

0.6682 

0.6703 

0.6724 

0.6745 

56 

0.6894 

0.6916 

0.6937 

0.6959 

0.6980 

0.7002 

55 

0.7155 

0.7177 

0.7199 

0.7221 

0.7243 

0.7265 

54 

0.7422 

0.7445 

0.7467 

0.7490 

0.7513 

0.7536 

53 

"  0.7696 

0.7720 

0.7743 

0.7766 

0.7789 

0.7813 

52 

•  0.7978 

0.8002 

0.8026 

0.8050 

0.8074 

0.8098 

51 

'•'  0.8268 

0.8292 

0.8317 

0.8341 

0.8366 

0.8391 

50 

I*  0.8566 

0.8591 

0.8617 

0.8642 

0.8667 

0.8693 

49 

I  >  0.8873 

0.8899 

0.8925 

0.8951 

0.8978 

0.9004 

48 

0.9190 

0.9217 

0.9244 

0.9271 

0.9298 

0.9325 

47 

0.9517 

0.9545 

0.9573 

0.9601 

0.9629 

0.9657 

46 

0.9856 

0.9884 

0.9913 

0.9942 

0.9971 

1.0000 

45 

25 

20 

15 

10 

5 

0 

Deg. 

Minutes. 


NATURAL  COTANGENT. 


212 


TRIGONOMETRICAL   FUNCTIONS. 
NATURAL  TANGENT. 


Minutes. 

Deg. 

0 

5 

10 

15 

20 

25 

30 

45 

1.0000 

1.0029 

1.0058 

1.0088 

1.0117 

1.0146 

1.0176 

46 

1.0355 

1.0385 

1.0416 

1.0446 

1.0477 

1.0507 

1.0538 

47 

1.0724 

1.0755 

1.0786 

10818 

1.0850 

1.0881 

1.0913 

48 

11106 

1.1139 

1.1171 

1.1204 

1.1237 

1.1270 

1.1303 

49 

1.1504 

1.1537 

1.1571 

1.1606 

1.1640 

1.1674 

1.1708 

50 

1.1917 

1.1953 

1.1988 

1.2024 

1.2059 

1.2095 

1.2131 

51 

1.2349 

1.2386 

1.2423 

1.2460 

1.2497 

1.2534 

1.2572 

52 

1.2799 

1.2838 

1.2876 

1.2913 

1.2954 

1.2993 

1.3032 

53 

1.3270 

1.3311 

1.3351 

1.3302 

1.3432 

1.3472 

1.3514 

54 

1.3764 

1.3806 

1.3848 

1.3891 

1.39^4 

1.3976 

1.4019 

55 

1.4281 

1.4326 

1.4370 

1.4415 

1.4460 

1.4505 

1.4550 

56 

1.4826 

1.4872 

1.4919 

1.4966 

1.5013 

1.5061 

1.5108 

57 

1.5399 

1.5448 

1.5497 

1.5547 

1.5597 

1.5647 

1.5697 

58 

1.6003 

1.6055 

1.6107 

1.6160 

1.6212 

1.6265 

1.6318 

59 

1.6643 

1.6698 

1.6753 

1.6808 

1.6864 

1.6920 

1.6976 

60 

1.7320 

1.7379 

1.7437 

1.7496 

1.7556 

1.7615 

1.7675 

61 

1.8040 

1.8102 

1.8165 

1.8228 

1.8291 

1.8354 

1.8418 

62 

1.8807 

1.8873 

1.8940 

1.9007 

1.9074 

1.9142 

1.9210 

63 

1.9626 

1.9697 

1.9768 

1.9840 

1.9912 

1.9984 

2.0057 

64 

2.0503 

2.0579 

2.0655 

2.0732 

2.0809 

2.0887 

2.0965 

65 

2.1445 

2.1527 

2.1609 

2.1692 

2.1775 

2.1859 

2.1943 

66 

2.2460 

2.2549 

2.2637 

2.2727 

2.2817 

2.2907 

2.2998 

67 

2.3558 

2.3654 

2.3750 

2.3847 

2.3945 

2.4043 

2.4142 

68 

2.4751 

2.4855 

2.4960 

2.5065 

2.5171 

2.5279 

2.5386 

69 

2.6051 

2.6165 

2.6279 

2.6394 

2.6511 

2.6628 

2.6746 

70 

2.7475 

2.7600 

2.7725 

2.7852 

2.7980 

2.8109 

2.8239 

71 

2.9042 

2.9180 

2.9319 

2.9456 

2.9600 

2.9743 

2.9886 

72 

3.0777 

3.0930 

3.1084 

3.1240 

3.1397 

3.1556 

3.1716 

73 

3.2708 

3.2879 

3.3052 

3.3226 

3.3402 

3.3580 

3.3759 

74 

3.4874 

3.5067 

3.5201 

3.5457 

3.5656 

3.5856 

3.6059 

75 

3.7320 

3.7539 

3.7760 

3.7983 

3.8208 

3.8436 

3.8667 

76 

4.0108 

4.C358 

4.0611 

4.0867 

4.1126 

4.1388 

4.1653 

77 

4.3315 

4.3604 

4.3897 

4.4194 

4.4494 

4.4799 

4.5107 

78 

4.7046 

4.7385 

4.7729 

4.8077 

4.8430 

4.8788 

4.9152 

79 

5.1445 

5.1848 

5.2257 

5.2671 

5.3093 

5.3521 

5.3955 

80 

5.6713 

5.7199 

5.7694 

5.8197 

5.8708 

5.9228 

5.9758 

81 

6.3137 

6.3737 

6.4348 

6.4971 

6.5605 

6.6252 

6.6912 

82 

7.1154 

7.1912 

7.2687 

7.3479 

7.4287 

7.5113 

7.5957 

83 

8.1443 

8.2434 

8.3450 

8.4490 

8.5555 

8.6648 

8.7769 

84 

9.5144 

9.6493 

9.7S82 

9.9310 

10.0780 

10.2290 

10.3850 

85 

11.4300 

11.6250 

11.8260 

12.0350 

12.2510 

12.4740 

12.7060 

86 

14.5010 

14.6060 

14.9240 

15.2570 

15.6050 

15.9690 

16.3500 

87 

19.0810 

19.6270 

20.2060 

20.8190 

21.4700 

22.1640 

22.9040 

88 

28.6360 

29.8820 

31.2420 

32.7300 

34.3680 

36.1780 

38.1880 

89 

57.2900 

62.4990 

68.7500 

76.3900 

85.9480 

98.2180 

114.5900 

60 

55 

50 

45 

40 

35 

30 

Deg. 

Minutes. 

NATURAL  COTANGENT. 


TBIGONOMETKICAL   FUNCTIONS. 
NATURAL  TANGENT. 


Minutes. 


Deg. 

35 

40 

45 

50 

55 

60 

1.0206 

1.0235 

1.0265 

1.0295 

1.0325 

1.0355 

44 

1.05G8 

1.0590 

1.0630 

1.0661 

1.0692 

1.0724 

43 

1.0945 

1.0977 

1.1009 

1.1041 

1.1074 

1.1106 

42 

1.1336 

1.1369 

1.1403 

1.1436 

1.1470 

1.1504 

41 

1.1743 

1.1778 

1.1812 

1.1847 

1.1882 

1.1917 

40 

1.2167 

1.2203 

1.2239 

1.2276 

1.2312 

1.2349 

39 

1.2609 

1.2647 

1.2685 

1.2723 

1.2761 

1.2799 

38 

1.3071 

1.3111 

1.3151 

1.3190 

1.3230 

1.3270 

37 

1.3555 

1.3597 

1.3638 

1.3680 

1.3722 

1-3764 

36 

1.4063 

1.4106 

1.4150 

1.4193 

1.4237 

1.4281 

35 

1.4595 

1.4641 

1.4687 

1.4733 

1.4779 

1.4826 

34 

1.5156 

1.5204 

1.5252 

1.5301 

1.5350 

1.5399 

33 

1.5747 

1.5798 

1.5849 

1.5900 

1.5952 

1.6003 

32 

1.6372 

1.6426 

1.6479 

1.6534 

1.6588 

1.6643 

31 

1.7033 

1.7090 

1.7147 

1.7205 

1.7263 

1.7320 

30 

1.7735 

1.7795 

1.7856 

1.7917 

1.7979 

1.8040 

29 

1.8482 

1.8546 

1.8611 

1.8676 

1.8741 

1.8807 

28 

1.9278 

1.9347 

1.9416 

1.9486 

1.9556 

1.9626 

27 

2.0130 

2.0204 

2.0278 

2.0353 

2.0428 

2.0503 

26 

2.1044 

2.1123 

2.1203 

2.1283 

2.1364 

2.1445 

25 

2.2028 

2.2113 

2.2199 

2.2286 

2.2373 

2.2460 

24 

2.3090 

2.3183 

2.3276 

2.3369 

2.3464 

2.3558 

23 

2.4242 

2.4342 

2.4443 

2.4545 

2.4648 

2.4751 

22 

2.5495 

2.5605 

2.5715 

2.5826 

2.5938 

2.6051 

21 

2.6865 

2.6985 

2.7106 

2.7228 

2.7351 

2.7475 

20 

2.8370 

2.8502 

2.8636 

2.8770 

2.8905 

2.9042 

19 

3.C032 

3.0178 

3.0326 

3.0475 

3.0625 

3.0777 

18 

3.1877 

3.2041 

3.2205 

3.2371 

3.2539 

3.2708 

17 

3.3941 

3.4124 

3.4308 

3.4495 

3.4684 

3.4874 

16 

3.6264 

3.6471 

3.6680 

3.6891 

3,7105 

3.7320 

15 

3.8900 

3.9136 

3.9375 

3.9616 

3.9861 

4.0108 

14 

4.1921 

4.2193 

4.2468 

4.2747 

4.3029 

4.3315 

13 

4.5420 

4.5736 

4.6057 

4.6382 

4.6712 

4.7046 

12 

4.9520 

4.9894 

5.0273 

5.0658 

5.1049 

5,1445 

11 

5.4397 

5.4845 

5.5301 

5.5764 

5.6234 

5.6713 

10 

6,0296 

6.0844 

6.1402 

6.1970 

6.2549 

6.3137 

9 

6.7584 

6.8269 

6.8969 

6.9682 

7.0410 

7.1154 

8 

7.6821 

7.7703 

7.8606 

7.9530 

8.0476 

8.1443 

7 

8.8918 

9.0098 

9.1309 

9.2553 

9.3831 

9.5144 

6 

10.5460 

10.7120 

10.8830 

11.0590 

11.2420 

11.4300 

5 

12.9470 

13.1970 

13.4570 

13.7270 

14.0080 

14.3010 

4 

16.7500 

17.1690 

17.6110 

18.0750 

18.5640 

19.0810 

3 

23.6940 

24.5420 

25.4520 

26.4320 

27.4900 

28.6360 

2 

40.4360 

42.9640 

45.8290 

49.1040 

52.8820 

57.2900 

1 

137.5100 

171.8800 

229.1800 

343.7700 

687.5500 

0 

25 

20 

15 

10 

5 

0 

Deg. 

Minutes. 


NATURAL  COTANGENT. 


214 


TRIGONOMETRICAL   FUNCTIONS. 
NATURAL  SECANT. 


Minutes. 

Deg. 

0 

5 

10 

15 

20 

25 

30 

0 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1 

1.0001 

1.0002 

1.0002 

1.00C2 

1.0003 

1.0003 

1.0003 

2 

1.0006 

1.0007 

1.0007 

1.0008 

1.0008 

1.0009 

1.0009 

3 

1.0014 

1.0014 

1.0015 

1.0016 

1.0017 

1.0018 

1.0019 

4 

1.0021 

1.0025 

1.0021 

1.0027 

1.0023 

1.0030 

1.0031 

5 

1.0038 

1.0039 

1.0041 

1.0042 

1.0043 

1.0045 

1.0046 

6 

1.0055 

1.0057 

1.0058 

l.OOGO 

1.0061 

1.0063 

1.0065 

7 

1.0075 

1.0077 

1.0079 

1.0080 

1.0082 

1.0084 

1.0086 

8 

1.0098 

1.0  LOO 

1.0102 

1.0104 

1.0107 

1.0109 

1.0111 

9 

1.0125 

1.0127 

1.0129 

1.0132 

1.0134 

1.0136 

1.0139 

10 

1.0154 

1.0157 

10159 

1.0162 

1.0165 

1.0167 

1.0170 

11 

1.0187 

1.0190 

1.0193 

1.0196 

1.0199 

1.0202 

1.0205 

12 

1  0223 

1.022ii 

1.0229 

1.0233 

1.02:56 

1.0239 

1.0243 

13 

1.0263 

1.0266 

1.0270 

1.0274 

1.0277 

1.0280 

1.0284 

14 

1.0306 

1.0310 

1.0314 

1.0317 

1.0321 

1.0325 

1.0329 

15 

1.0353 

1.0357 

1.0361 

1.0365 

1.0369 

1.0373 

1.0377 

16 

1.0403 

1.0407 

1.0412 

1.0416 

1.0420 

1.0425 

1.0429 

17 

1.0457 

1.0461 

1.0466 

1.0471 

1.0476 

1.0480 

1.0485 

18 

1.0515 

1.0520 

1.0525 

1.0530 

1.0535 

1.0540 

1.0545 

19 

1.0577 

1.0581 

1.0587 

1.0592 

1.0598 

1.0G03 

1.0608 

20 

1.0642 

1.0647 

1.0653 

1.0659 

1.0664 

1.0ti70 

1.0676 

21 

1.0711 

1.0717 

1.0723 

1.0729 

1.0736 

1.0742 

1.0748 

22 

1.0785 

1.0792 

1.0798 

1.0804 

1.0811 

1.0817 

1.0824 

23 

1.0864 

1.0870 

1.0877 

1.0884 

1.0891 

1.0897 

1.0904 

24 

1.0946 

1.0953 

1.0961 

1.0968 

1.0975 

1.09S2 

1.0989 

25 

1.1034 

1.1041 

1.1049 

1.1056 

1.1064 

1.1072 

1.1079 

20 

1.1126 

1.1134 

1.1142 

1.1150 

1.1158 

1.11  (56 

1.1174 

2T 

1.1223 

1.1231 

1.1240 

1.1248 

1.1257 

1.1265 

1.1274 

28 

1.1326 

1.1334 

1.1343 

1.1352 

1.1361 

1.1370 

1.1379 

29 

1.1433 

1.1443 

1.1452 

1.1461 

1.1471 

1,1480 

1.1489 

30 

1.1547 

1.1557 

1-1566 

1.1576 

1.1586 

1.1596 

1.1606 

31 

1.1666 

1.1676 

1.1687 

1.1697 

1.1707 

1.1718 

1.1728 

32 

1.1792 

1.1802 

1.1830 

1.1824 

1.1835 

1.1846 

1.1857 

33 

1.1923 

1.1935 

1.1946 

1.1958 

1.1969 

1.1980 

1.1992 

34 

1.2002 

1.2074 

1.2068 

1.2098 

1.2110 

1.2122 

1.2134 

35 

1.2208 

1.2220 

1.2233 

1.2245 

1.2258 

1.2270 

1.2283 

36 

1.2361 

1.2374 

1.2387 

1.2400 

1.2413 

1,2427 

1.2440 

37 

1.2521 

1.2535 

1.2549 

1.2563 

1.2577 

1.2591 

1.2605 

38 

1.2690 

1.2705 

1.2719 

1.2734 

1.2748 

1.2763 

1.2778 

39 

1.2867 

1.2883 

1.2898 

12913 

1,2929 

1.2944 

1.29GO 

40 

1.3054 

1.3070 

4.3086 

1.3102 

1.3118 

1.3134 

1.3151 

41 

1.3250 

1.3267 

1.3284 

1.3301 

1.3318 

1.3335 

1.3352 

42 

1.3456 

1.3474 

1.3492 

1.3509 

1.3507 

1.3540 

1.3563 

43 

1.3673 

1.3692 

1.3710 

3,3729 

1,3748 

1.3767 

1.3786 

44 

1.3902 

1.3921 

1.3941 

1.3960 

1.3980 

1.4000 

1.4020 

60 

55 

50 

45 

40 

35 

30 

Deg. 

Minutes. 

NATURAL  COSECANT. 


TRIGONOMETRICAL   FUNCTIONS. 
NATURAL  SECANT. 


Minutes. 


L>eg. 

35 

40 

45 

50 

55 

60 

1.0000 

1.0001 

1.0001 

1.0001 

1.0001 

1.0001 

89 

1.0004 

1.0004 

1.0005 

1.0005 

1.0005 

1.0(»OG 

88 

1.0010 

1.0011 

1.0011 

1.0012 

1.0013 

1.0014 

87 

1.0019 

1.0020 

1.0021 

1.0022 

1.0023 

1.0024 

86 

1.0032 

1.0033 

1.0034 

1.0036 

1.0037 

1.0038 

85 

1.0048 

1.0049 

1.0050 

1.0052 

1.0053 

1  .0055 

84 

1.00G6 

1.0008 

1.0070 

1.0071 

1.0073 

1.0075 

83 

1.0088 

1.0090 

1.0092 

1.0094 

1.0096 

1.0098 

82 

1.0113 

1.0115 

1.0118 

1.0120 

1.0122 

1.0125 

81 

1.0141 

1.0145 

1.0146 

1.0149 

1.0152 

1.0154 

80 

1.0173 

10176 

1.0179 

1.0181 

1.0184 

1.0187 

79 

1.0208 

1.0211 

1.0214 

1.0217 

1.0220 

1.0223 

78 

1.0246 

1.0249 

1.0253 

1.0256 

1.0260 

1.0263 

77 

1.0288 

1.0291 

1.0295 

1.0298 

1.0302 

1.0306 

76 

1.0333 

1.0337 

1.0341 

l.<  345 

1.0349 

1.0353 

75 

1.03S2 

1.0386 

1.0390 

1.0394 

1.0399 

1.0403 

74 

1.0434 

1.0438 

1.0443 

1.0448 

1.0452 

1.0457 

73 

1.0490 

1.0495 

1.0500 

1.0505 

1.0510 

1.0515 

72 

1.0550 

1.0555 

1.0560 

1.0565 

1.0571 

1.0577 

71 

1.0644 

1.0619 

1.0625 

1.0630 

1.0636 

1.0642 

70 

1.0082 

1.0688 

1.0694 

1.0699 

1.0705 

1.0711 

69 

1.0754 

1.0760 

1.0766 

1.0773 

1.0779 

1.0785 

68 

1.0830 

1.0837 

1.0844 

1.0850 

1.0857 

1.0864 

67 

1.0911 

1.0918 

1.0925 

1.0932 

1.0939 

1.0946 

66 

1.0997 

1.1004 

1.1011 

1.1019 

1.1026 

1.1034 

65 

1.1087 

1.1095 

1.1102 

1.1110 

1.1118 

1.1126 

64 

1.1182 

1.1190 

1.1198 

1.1207 

1.1215 

1.1223 

63 

1.1282 

1.1291 

1.1299 

1.1308 

2.1317 

1.1326 

62 

1.1388 

1.1397 

1.1406 

1.1415 

1.1424 

1.1433 

61 

1.1499 

1.1508 

1.1518 

1.1528 

1.1537 

1.1547 

60 

1.1616 

1.1626 

1.1636 

1.1646 

1.1656 

1.1666 

59 

1.1739 

1.1749 

1.1760 

1.1770 

1.1781 

1.1792 

58 

1.1808 

1.1879 

1.1819 

1.1901 

1.1912 

1.1923 

57 

1.2004 

1.2015 

1.2027 

1.2039 

1.2050 

1.2062 

56 

1.2146 

1.2158 

1.2171 

1.2183 

1.2195 

1.2208 

55 

1.2296 

1.2309 

1.2322 

1.2335 

1.2348 

1.2361 

54 

1.2453 

1.2467 

1.2480 

1.2494 

1.2508 

1.2521 

53 

1.2619 

1.2633 

1.2647 

1.2661 

1.2676 

1.2690 

52 

1.2793 

1.2807 

1.2822 

1.2837 

1.2852 

1.2867 

51 

1.2975 

1.2991 

1.3006 

1.3022 

1.3038 

1.3054 

50 

1.3167 

1.3184 

1.3200 

1.3217 

1.3233 

1.3250 

49 

1.3369 

1.3386 

1.3404 

1.3421 

1.3439 

1.3450 

48 

1.3581 

1.3GOO 

1.3618 

1.3636 

1.3655 

1.3073 

47 

1.3805 

1.3824 

1.3843 

1.3863 

1.3882 

1.3902 

46 

1.4040 

1.4056 

1.4081 

1.4101 

1.4122 

1.4142 

45 

25 

20 

15 

10 

5 

0 

Dee 

Minutes. 


NATURAL  COSECANT. 


216 


TRIGONOMETRICAL   FUNCTIONS. 
NATURAL  SECANT. 


Minutes. 

Deg. 

0 

5 

10 

15 

20 

25 

30 

45 

1.4142 

1.4163 

1.4183 

1.4204 

1.4225 

1.424G 

1.4267 

46 

1.4395 

1.4417 

1.4439 

1.44G1 

1.4483 

1.4505 

1.4527 

47 

1.46G3 

1.4G86 

1.4709 

1.4732 

1.4755 

1.4778 

1.4802 

48 

14945 

1.4969 

1.4993 

1.5018 

1.5042 

1.50G7 

1.5092 

49 

1.5242 

1.5268 

1.5294 

1.5319 

1.5345 

15371 

1.5398 

50 

1.5557 

1.5584 

1.5011 

1.5639 

1.5GGG 

1.5G94 

1.5721 

51 

1.5890 

1.5919 

1.5947 

1.5976 

1.6005 

l.GO:54 

1.6064 

52 

1.6243 

1.6273 

1.6303 

1.6334 

1.6365 

1.6396 

1.6427 

53 

1.6616 

1.6648 

1.GG81 

1.6713 

1.6746 

1.6779 

1.6812 

54 

1.7013 

1.7047 

1.7081 

1.7116 

1.7151 

1.7185 

1.7220 

55 

1.7434 

1.7471 

1.7507 

1.7544 

1.7581 

1.7018 

1.7655 

56 

1.7883 

1.7921 

1.7960 

1.7999 

1.8039 

1.8078 

1.8118 

57 

1.8361 

1.8402 

1.8443 

1.8485 

1.8527 

1.8569 

1.8G11 

58 

1.8871 

1.8915 

1.8959 

1.9004 

1.9048 

1.9093 

1.9139 

59 

1.9416 

1.9463 

1.9510 

1.9558 

1.9606 

1.9C54 

1.9703 

60 

2.0000 

2.0050 

2.0102 

2.0152 

2.0204 

2.0256 

2.0308 

61 

2.0(527 

2.0681 

2.0735 

2.0790 

2.0846 

2.0901 

2.0957 

62 

2.1300 

2.1359 

2.1418 

2.1477 

2.1536 

2.1596 

2.1657 

63 

2.2027 

2.2090 

2.2153 

22217 

2.2282 

2.2346 

2.2411 

64 

2.2812 

2.2880 

2.2949 

2.3018 

2.3087 

2.3158 

2.3228 

65 

2.3662 

2.3736 

2.3811 

2.3886 

2.3961 

2.4037 

2.4114 

66 

2.4586 

2.4G6G 

2.4748 

2.4829 

2.4912 

2.4995 

2.5078 

67 

2.5593 

2.5G81 

2.5770 

2.5859 

2.5949 

2.G040 

2.6131 

68 

2.G695 

2.G791 

2.6888 

2.6986 

2.7085 

2.7185 

2.7285 

69 

2.7904 

2.8010 

2.8117 

2.8225 

2.8334 

2.8444 

2.8554 

70 

2.9238 

2.9355 

2.9474 

2.9593 

2.9713 

2.9835 

2.9957 

71 

3.0715 

3.0S4G 

3.0977 

3.1110 

3.1244 

3.1379 

3.1515 

72 

3.23G1 

3.250G 

3.2G53 

3.2801 

3.2951 

3.3102 

3.3255 

73 

3.4203 

3.4366 

3.4532 

3.4G97 

3.4867 

3.5037 

3.5209 

74 

3.6276 

3.6464 

3.GG51 

3.68-10 

3.7031 

3.7224 

3.7420 

*  75 

3.8G37 

3.8848 

3.9061 

3.9277 

3.9495 

3.9716 

3.9939 

76 

4.1330 

4.1578 

4.1824 

4.2072 

4.2324 

4,2579 

4.2836 

77 

4.4454 

4.4736 

4.5021 

4.5331 

4.5604 

4.5901 

4.6202 

78 

4.8097 

4.8429 

4.8765 

4.9106 

4.9452 

4.9802 

5.0158 

79 

5.2408 

5.2803 

5.3205 

5.3G12 

5.4020 

5.4447 

5.4874 

80 

5.7588 

5.80G7 

5.8554 

5.9049 

5.9554 

5.99G3 

6.0588 

81 

6.3924 

6.4517 

6.5121 

6.5736 

6.6363 

6.7003 

6.7655 

82 

7.1853 

7.2604 

7.3372 

7.4156 

7.4957 

7.5776 

7.6613 

83 

8.2055 

8.3C39 

8.4046 

8.5079 

8.6138 

8.7223 

8.8337 

84 

9.5608 

9.7010 

9.^391 

9.9812 

10.1270 

10.2780 

10.4330 

85 

11.4740 

11.6680 

11.8680 

12.07GO 

12.2910 

12.5140 

12.7450 

86 

14.3350 

14.6400 

14.9580 

15.2900 

15.6370 

16.0000 

16.3800 

87 

19.1070 

19.G530 

20.23(10 

20.8430 

21.4940 

22.18GO 

22.9250 

88 

28.6540 

29.8990 

31.2570 

32.7450 

34.3820 

36.1910 

38.2010 

89 

57.2990 

62.5070 

68.7570 

76.3960 

85.9460 

98.2230 

114.5900 

GO 

55 

50 

45 

40 

35 

30 

Deg 

Minutes. 

NATURAL  COSECANT. 


TRIGONOMETRICAL   FUNCTIONS. 
NATURAL  SECANT. 


Minutes. 


35 

40 

45 

50 

55 

60 

1.4288 

1.4310 

1.4331 

1.4352 

1.4374 

1.4395 

1.4550 

1.4572 

1.4595 

1.4617 

1.4640 

1.4663 

1.4825 

1.4849 

1.4873 

1.4897 

1.4921 

1.4945 

1.511G 

1.5141 

1.5166 

1.5192 

1.5217 

1.5242 

1.5424 

1.5450 

1.5477 

1.5503 

1.5530 

1.5557 

1.5749 

1.5777 

1.5805 

1.5833 

1.5862 

1.5890 

1.G093 

1.6123 

1.6153 

1.6182 

1.6212 

1.6243 

1.6458 

1.6489 

1.6521 

1.6552 

1.6584 

1.6616 

1.6845 

1.6878 

1.6912 

1.6945 

1.6979 

1.7013 

1.7256 

1.7291 

1.7327 

1.7362 

1.7398 

1-7434 

1.7693 

1.7730 

1.7768 

1.7806 

1.7844 

1.7883 

1.8158 

1.8198 

1.8238 

1.8279 

1.8320 

1.8361 

1.8654 

1.8G97 

1.8740 

1.8783 

1.8827 

1.8871 

1.9184 

1.9230 

1.9276 

1.9322 

1.9369 

1.9416 

1.9752 

1.9801 

1.9850 

1.9900 

1.9950 

2.0000 

2.0360 

2.0413 

2.0466 

2.0519 

2.0573 

2.0627 

2.1014 

2.1070 

2.1127 

2.1185 

2.1242 

2.1300 

2.1717 

2.1778 

2.1840 

2.1902 

2.1964 

2.2027 

2.2477 

2.2543 

2.2610 

2.2676 

2.2744 

2.2812 

2.3299 

2.3371 

2.3443 

2.3515 

2.3588 

2.3662 

2.4191 

2.4269 

2.4347 

2.4426 

2.4506 

2.4586 

2.5163 

2.5247 

2.5333 

2.5419 

2.5506 

2.5593 

2.G223 

2.6316 

2.6410 

2.6504 

2.6599 

2.6695 

2.7386 

2.7488 

2.7591 

2.7694 

2.7799 

2.7904 

2.8666 

2.8778 

2.8892 

2.9006 

2.9122 

2.9338 

3.0081 

3.0206 

3.0331 

3.0458 

3.0586 

3.0715 

3.1653 

3.1792 

3.1932 

3.2074 

3.2216 

3.2361 

3.3409 

3.3565 

3.3722 

3.3881 

3.4041 

3.4203 

3.53S3 

3.5559 

3.5736 

3.5915 

3.6096 

3.6279 

3.7617 

3.7816 

3.8018 

3.8222 

3.8428 

3.8637 

4.0165 

4.0394 

4.0625 

4.0859 

4.1090 

4.1336 

4.3098 

4.3362 

4.3630 

4.3901 

4.4176 

4.4454 

4.6507 

4.6817 

4.7130 

4.7448 

4.7770 

4.8097 

5.0520 

5.0886 

5.1258 

5,1636 

5.2019 

5.2408 

5.5308 

5.5749 

5.6197 

5.6653 

5.7117 

5.7588 

6.1120 

6.166  1 

6,2211 

6.2772 

6.3343 

6.3924 

6.8320 

6,8998 

6.9690 

7,0390 

7.1117 

7.1853 

7.7469 

7.8344 

7.9240 

7.9971 

8.1094 

8.2055 

8.9479 

9.0651 

9.1855 

9.3092 

9.4362 

9.5668 

10.5930 

10.7580 

10.9290 

11.1040 

11.2080 

11,4740 

12.9850 

13.2350 

13.4940 

13.7630 

14.0430 

14.3350 

16.7790 

17.1980 

17.6390 

18.1030 

18.5910 

19.1070 

23.7160 

24,5620 

25.4710 

26.1500 

27.5080 

28.6540 

39.9780 

42,9760 

45.8400 

49.1140 

52.8910 

57.2990 

137.5100 

171.8900 

229.1800 

343.7700 

687.5500 

00 

25 

20 

15 

10 

5 

0 

Minutes. 


NATURAL  COSECANT, 


218      CIRCUMFERENCE,  AREA,  AND  CUBIC  CONTENTS  OF  CIRCLES. 


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CIRCUMFERENCE,  AREA,  AND  CUBIC  CONTENTS  OF  CIRCLES.        221 

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222      CIRCUMFERENCE,  AREA,  AND  CUBIC  CONTENTS  OF  CIRCLES 


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CIRCUMFERENCE,  AREA,  AND  CUBIC  CONTENTS  OF  CIRCLES. 


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221 


SPECIFIC  GRAVITIES  OF  MATERIALS. 


SPECIFIC  GRAVITIES  OF  MATERIALS. 


Weight  of 

a  cubic  foot 

in  Ibs. 


GASES  at  32°  Fahr.,  and  under  the  pressure  of  on 
atmosphere  of  2116.4  Ibs.  on  the  square  foot: 
Air  

e 

0.080728 

Carbonic  acid  0.12344 

Hydrogen  0.005592 

Oxygen  0.089256 

Nitrogen  0  078596 

Steam  (ideal)  0.05022 

^Ether  vapor  (ideal)  .                                    02093 

Bisulphuret-of-carbon  vapour  (i 

ieal)  0  2137 

0  079R 

LIQUIDS  at  32°  Fahr.  (except  water, 
which  is  taken  at  39°.4  Fahr.): 
Water,  pure,  at  39°.4  

Weight  of  a 
cubic  foot  in  Ibs. 
avoirdupois. 

Specific 
gravity,  pure 
water  =  I. 

62.425 
64.05 
49.38 
57.18 
44.70 
848.75 
52.94 
58.68 
57.12 
57.62 
54.31 
54.81 

187.3 
125  to  135 
112 
117  to  174 
120 
100 
77.  4  to  89.  9 
62.43  to  103.  6 
162.3 
164.2 

1.000 
1.026 
0.791 
0.916 
0.716 
13.596 
0.848 
0.940 
0.915 
0.923 
0.870 
0.878 

3.00 
2  to  2.  167 
1.8 
1.87  to  2.  78 
1.92 
1.602 
1.24  to  1.44 
1.00  to  1.66 
2.6 
2.63 

"        sea  ordinary  

Alcohol  pure 

proof  spirit  

^Ether                 .                

Mercury  

Naphtha 

Oil,  linseed  

"     olive 

"     whale  

"     of  turpentine 

SOLID  MINERAL  SUBSTANCES,  non- 
metallic: 
Basalt  

Brick  

Brickwork  

Chalk                          .            .... 

Clay  

Coal  anthracite  

"      bituminous              . 

Coke      ,  

Felspar  

Flint.. 

SPECIFIC  GRAVITIES  OF  MATERIALS. 


225 


SOLID  MINERAL  SUBSTANCES  —  con 

Weight  of  a 
cubic  foot  in  Ibs. 
avoirdupois. 

Specific 
gravity,  pure 
water  =  1. 

tinued: 
Glass  crown  average  

156 

2  5 

11      flint  

187 

3.0 

"      green  

169 

2.7 

"      plate       

169 

2  7 

Granite  

164  to  172 

2.63  to  2.76 

Gypsum  

143.6 

2.3 

Limestone,  (including  marble)... 
magnesian  

169  to  175 

178 

2.  7  to  2.8 

2.86 

Marl                    '    ...    . 

100  to  119 

1  6  to  1  9 

Masonry  

116  to  144 

1.85  to  2  3 

Mortar  

109 

1.75 

Mud  

102 

1  63 

Quartz  

165 

2  65 

Sand  (damp)        .... 

118 

1  9 

"      (dry).  . 

88.6 

1  42 

Sandstone  average 

144 

2  3 

"          various  kinds   . 

130  to  157 

2  08  to  2  52 

Shale  

162 

2  6 

Slate  

175  to  181 

2  8  to  2  9 

Trap... 

170 

2  72 

METALS,  solid: 
Brass,  cast  

487  to  524.4 

7  8  to  8  4 

"      wire. 

533 

8  54 

Bronze  

524 

8  4 

Copper,  cast  

537 

8  6 

"       sheet     

549 

8  8 

"        hammered  

556 

8  9 

Gold  

1186  to  1224 

19  to  19  6 

Iron,  cast  various  

434  to  456 

6  95  to  7  3 

average  

444 

7  11 

Iron,  wrought  various 

474  to  487 

7  6  to  7  8 

average  

480 

7  69 

Lead  

712 

114- 

Platinum  

1311  to  1373 

21  to  22 

Silver  

655 

10   ^ 

Steel  

487  to  493 

7  8  to  7  9 

Tin  

456  to  468 

7  3  to  7  5 

Zinc  

424  to  449 

6  8  to  7  2 

TIMBER:* 
Ash  

47 

0   7^3 

Bamboo  

25 

0  4 

Beech  

43 

0  69 

15 

226 


SPECIFIC  GRAVITIES  OF  MATERIALS. 


TIMBER  :*  —  continued. 
Birch  .    .. 

Weight  of  a 
cubic  foot  in  Ibs. 
avoirdupois. 

Specific 
gravity,  pure 
water  =  1. 

44.4 
52.5 
60 
65.3 
56.2 
30.4 
33.4 
36.2 
74.5 
34 
30  to  44 
30  to  44 
29 
31  to  35 
62.5 
57 
54 
47 
47 
57 
42  to  63 

41  to  83 
44 
35 
53 
49 
57 
43  to  62 
54 
36 
60 
37 
41  to  55 
61 
62  to  66 
62.5 
25 
50 

0.711 
0.843 
0.96 
1.046 
0.9 
0.486 
0.535 
0.579 
1.193 
0.544 
0.48  to  0.7 
0.48  to  0.7 
0.46 
0.5  to  0.56 
1.001 
0.91 
0.86 
0.76 
0.76 
0.92 
0.675  to  1.01 

0.65  to  1.33 
0.71 
0.56 
0.85 
0.79 
0.92 
0.69  to  0.99 
0.87 
0.58 
0.96 
0.59 
0.66  to  0.88 
0.98 
0.99  to  1.06 
1.001 
0.4 
0.8 

Blue-gum  

Box  

Bullet-  tree  

Cabacalli  

Cedar  of  Lebanon  

Chestnut  

Cowrie  

Ebony,  West  Indian  

Elm  

Fir,  red  pine  

"     spruce  

"     American  yellow  pine  

Greenhart  

Haw  thorn  

Ha*zel  

Holly  

Hornbeam            

Laburnum  

Lancewood  

Larch.     (See  "fir".) 
Lignum-vitae  

Locust  

Mahogany,  Honduras  

Spanish  . 

Maple  

Mora  

Oak   European 

"     American  red  

Poon  

Saul 

Sy  cam  ore  

Teak  Indian  

11      African  

Tonka  

Water-gum  

Willow  

Yew  

*The  timber  in  every  case  is  supposed  to  be  dry. 


WEIGHT  OF  A  SUPERFICIAL  INCH.  ETC. 


227 


WEIGHT  OF  A  SUPERFICIAL  INCH  OF  WROUGHT  AND 
CAST  IRON. 

(From  one-sixteenth  to  one-inch  thickness.) 


Thickness  in 
inches. 

WROUGHT  IRON. 
Cubic  foot  =  480  Ibs. 

CAST  IRON. 
Cubic  foot  =  450  Ibs. 

Weight  in  Ibs. 

Weight  in  Ibs. 

A 

0.017356 

0.0163 

i 

0.0347 

0.0326 

A 

0.0520 

0.0489 

i 

0.0694 

0.0652 

T56 

0.0867 

0.0815 

1 

0.1041 

0.0978 

ft 

0.1214 

0.1141 

i 

0.1388 

0.1304 

T9* 

0.1562 

0.1467 

i 

0.1735 

0.1630 

« 

0.1909 

0.1793 

1 

0.2082 

0.1956 

n 

0.2256 

0.2119 

i 

0.2429 

0.2282 

it 

0.2603 

0.2445 

i 

0.2777 

0.2608 

228 


WEIGHT  PEE  SQUARE  FOOT  IH  POUNDS  AVOIRDUPOIS. 


WEIGHT  PER  SQUARE  FOOT  IN  POUNDS  AVOIRDUPOIS. 


Thickness  in 
inches. 

Wrought 
Iron. 

Cast  Iron. 

Copper, 
sheet. 

Lead. 

Zinc. 

480  Ibs.  per 
cubic  foot. 

450  Ibs.  per 
cubic  foot. 

549  Ibs.  per 
cubic  foot. 

712  Ibs.  per 
cubic  foot. 

436  Ibs.  per 
cubic  foot. 

TV 

2.50 

2.34 

2.86 

3.71 

2.27 

* 

5.00 

4.69 

5.72 

7.42 

4.54 

T36 

7.50 

7.03 

8.58 

11.12 

6.81 

i 

10.00 

9.37 

11.44 

14.83 

9.08 

ft 

12.50 

11.72 

14.30 

18.54 

11.35 

1 

15.00 

14.06 

17.16 

22.25 

13.62 

ft 

17.50 

16.41 

20.02 

25.96 

15.89 

» 

20.00 

18.75 

22.88 

29.66 

18.16 

ft 

22.50 

21.09 

25.74 

33.37 

20.43 

1 

25.00 

23.44 

28.60 

37.10 

22.70 

H 

27.50 

25.78 

31.46 

40.79 

24.97 

f 

30.00 

28.12 

34.32 

44.50 

27.24 

it 

32.50 

30.47 

37.18 

48.20 

29.51 

* 

35.00 

32.81 

40.04 

51.91 

31.78 

it 

37.50 

35.16 

42.90 

55.62 

34  05 

i 

40.00 

37.50 

45.75 

59.33 

36.33 

WEIGHT  OF  A  LINEAL  FOOT,  ETC. 


229 


WEIGHT  OF  A  LINEAL  FOOT  OF  FLAT  AND  SQUAKE 
BAR  IRON  IN  POUNDS  AVOIRDUPOIS. 

(480  pounds  per  cubic  foot.) 


Breadth  in 
inches. 

Thickness  in 
inches. 

Weight  in  Ibs. 

d 
5  2 
ll 

£ 

Thickness  in 
inches. 

Weight  in  Ibs. 

Breadth  in 
inches. 

Thickness  in 
inches. 

Weight  in  Ibs. 

i 

i 

0.104 

ij 

1 

5.000 

2J 

1 

1.875 

0.208 

H 

5.625 

| 

2.813 

1 

1 

0.208 

" 

if 

6.250 

" 

3.750 

0.416 

" 

if 

6.874 

" 

4.687 

" 

1 

0.832 

" 

if 

7.500 

M 

5.624 

1 

| 

0.312 

it 

4 

0.739 

« 

6.562 

I 

0.624 

i 

1.459 

11 

1 

7.500 

(( 

I 

0.937 

" 

I 

2.187 

" 

li 

8.437 

(( 

1.249 

•* 

2.916 

" 

H 

9.374 

II 

.| 

1.562 

<( 

1 

3.646 

rt 

If 

10.310 

(( 

j 

1.874 

•« 

i 

4.375 

" 

If 

11.250 

1 

i 

0.416 

" 

•§- 

5.103 

" 

If 

12.190 

" 

0.833 

•« 

5.833 

11 

If 

13.120 

<( 

-1 

1.249 

«' 

» 

6.562 

'• 

l| 

14.060 

M 

j 

1.667 

H 

u 

7.291 

11 

o 

15.000 

« 

-1 

2.089 

" 

if 

8.020 

H 

2J 

15.940 

« 

t 

2.500 

" 

if 

8.750 

11 

2i 

17.810 

11 

7 
F 

2.916 

" 

if 

9.478 

2} 

| 

1.041 

(t 

1 

3.333 

u 

it 

10.930 

2.089 

11 

i 

0.521 

2 

1 

0  833 

" 

| 

3.125 

| 

1.041 

" 

I 

1.667 

11 

1 

4.166 

(i 

f 

1.562 

'« 

3 
8" 

2.500 

" 

I 

5.208 

« 

2.Q89 

11 

| 

3.333 

" 

I 

6.250 

II 

2.603 

11 

4.166 

11 

i 

7.291 

<( 

3.124 

11 

£ 

5.000 

" 

8.333 

« 

3.646 

«c 

I 

5.833 

" 

li 

9.398 

<( 

1 

4.166 

" 

6.666 

11 

10.410 

ii 

1* 

4.687 

11 

If 

7.500 

" 

If 

11.460 

<( 

li 

5.728 

" 

l| 

8.333 

" 

IJ 

12.500 

i| 

J 

0.624 

11 

If 

9.156 

(1 

If 

13.540 

(i 

\ 

1.250 

u 

if 

10.000 

" 

If 

14.580 

11 

i 

1.875 

11 

If 

10.830 

•« 

If 

15.620 

ii 

2.500 

11 

If 

11.660 

11 

2 

16.660 

« 

3.125 

M 

li 

12.500 

" 

a 

17.710 

ii 

3.750 

M 

2 

13.330 

(( 

2i 

18.750 

ii 

4.375 

2i 

4 

0.937 

" 

2| 

20.820 

230 


WEIGHT  OF  A  LINEAL  FOOT,  ETC. 


Breadth  in 
inches. 

Thickness  in 
inches. 

Weight  in  Ibs. 

Breadth  in 
inches. 

Thickness  in 
inches. 

Weight  in  Ibs. 

Breadth  in 
inches. 

Thickness  in 
inches. 

Weight  in  Ibs. 

2J 

2f 

19.800 

3J 

2 

21.660 

4 

2 

26.660 

2J 

^ 

1.146 

i 

24.370 

" 

21 

30.000 

i 

2.292 

" 

2£ 

27.080 

" 

i 

33.330 

1 

^ 

3.437 

11 

2| 

29.790 

M 

2f 

36.660 

4 

| 

4.583 

" 

3 

32.500 

u 

3 

40.000 

1 

-| 

5.729 

41 

3i 

24.200 

M 

31 

43.330 

1 

J 

6.874 

3i 

i 

2.916 

" 

3* 

46.660 

1 

I 

8.020 

i 

5.833 

11 

3| 

50.000 

* 

1 

9.154 

u 

i 

8.750 

" 

4 

53.330 

«*' 

H 

10.310 

" 

11.660 

$ 

i 

3.541 

1 

u 

11.460 

11 

H 

14.580 

7.082 

' 

if 

12.600 

11 

U 

17  500 

" 

| 

10.620 

« 

i| 

13.750 

M 

if 

20.430 

M 

1 

14.160 

1 

i| 

14.900 

11 

2 

23.330 

M 

It 

16.800 

1 

ij 

16.030 

" 

2» 

26.250 

'• 

H 

21.330 

1 

1J 

17.190 

» 

2J 

29.160 

« 

If 

24.780 

' 

2 

18.330 

M 

2f 

32.080 

i 

2 

28.330 

« 

2J 

19.480 

" 

3 

35.000 

i 

2| 

31.870 

' 

i 

20.620 

M 

3| 

37.910 

1 

1 

35.410 

c 

1 

21.770 

» 

3! 

40.830 

1 

i 

38.950 

1 

2^ 

22.910 

3f 

3.125 

' 

3 

42.500 

1 

2| 

24.060 

j 

6.250 

' 

3J 

46.030 

1 

2| 

25.200 

11 

j 

9.375 

1 

3£ 

49.570 

3 

i 

2.500 

it 

i 

12.500 

( 

3| 

53.120 

' 

5.000 

" 

14 

15.620 

1 

4 

56.660 

1 

i 

7.500 

" 

1} 

18.750 

' 

« 

60.200 

1 

i 

10.000 

M 

if 

21.870 

IF 

i 

3.750 

1 

ia 

12.500 

" 

2 

25.000 

£ 

7.500 

1 

i? 

15.000 

" 

2* 

28.120 

y 

| 

11.250 

• 

if 

17.500 

«« 

2J 

31.250 

M 

1 

15.000 

".":  «  :, 

2 

20.000 

» 

2f 

34.370 

" 

it 

18.750 

1 

2} 

22.500 

" 

3 

37.500 

! 

9 

22.500 

1 

2J 

25.000 

M 

31 

40.620 

1 

if 

26.250 

' 

2| 

27.500 

" 

3^ 

43.750 

' 

2 

30.000 

• 

3 

30.000 

" 

3f 

46.860 

1 

^ 

33.750 

3| 

| 

2.708 

4 

i 

3.330 

1 

^ 

37.500 

5  416 

" 

i 

6.660 

< 

2f 

41.250 

'C  '"**;« 

|. 

8.124 

" 

f 

10.000 

f 

3 

45.000 

!£t  .**.] 

1 

10.830 

«* 

13.330 

« 

5i 

48.750 

..;_'  «;,. 

H 

13.500 

" 

11 

16.660 

1 

3J 

52.500 

4 

1| 

16.250 

M 

1£ 

20.000 

M 

si 

56.250 

t 

If 

18.950 

a 

If 

23.330 

N 

4 

60.000 

WEIGHT  OF  A  LINEAL  FOOT,  ETC. 


231 


£ 

•5  * 

•§1 

a/  G 
t,'1-1 

PQ 

Thickness  in 
inches. 

Weight  in  Ibs. 

a 

5s 

~Q-Z 
06  « 

P 

PQ 

Thickness  in 
inches. 

J5 

3f 
« 

Breadth  in 
inches. 

co  • 
CJ  <P 
g-g 

!s-S 

^ 

Weight  in  Ibs. 

4J 

4} 

63  .  750 

5i 

i 

8.753 

5| 

i 

4.788 

4£ 

67.500 

-1 

1-3.130 

i 

9.587 

4f 

1 

3.953 

I 

17.500 

( 

1 

14.370 

i 

7.910 

1 

H 

21.870 

i 

1 

19.160 

" 

i 

11.860 

1 

H 

26.250 

i 

1} 

23.950 

•« 

15.830 

1 

if 

30.620 

i 

li 

28.750 

ti 

I* 

19.760 

f 

2 

35.000 

1 

I 

33.540 

« 

li 

23.750 

IJ 

m 

39.370 

• 

2 

38.330 

< 

If 

27.700 

u 

2i 

43.750 

" 

2t 

43.120 

• 

2 

31.670 

1 

2| 

48.110 

c< 

2J 

47.910 

i 

2i 

35.620 

' 

3 

52.500 

M 

2| 

52.700 

1 

2i 

39.580 

1 

3J 

56.680 

( 

3 

57.500 

1 

2| 

43.540 

' 

3i 

61.250 

1 

3i 

62.300 

1 

3 

47.500 

1 

3| 

65.620 

4 

3i 

67.080 

1 

P 

51.460 

4 

4 

70.000 

( 

3f 

71.860 

M 

31 

55.410 

| 

4} 

74.370 

4 

4 

76.650 

(i 

3| 

59.370 

« 

4| 

78.750 

4 

H 

81.450 

•« 

4 

63.330 

4 

4f 

83.110 

1 

4J 

86.240 

« 

4} 

67.290 

' 

5 

87.500 

' 

4f 

91.030 

M 

4 

71.250 

! 

5J 

91.860 

4 

5 

95.820 

« 

4| 

75.200 

5J 

J 

4.587 

4 

5t 

100.600 

5 

1 

4.166 

i 

9.164 

( 

5} 

105.400 

M 

i 

8.330 

44 

1 

13.750 

4 

5| 

119.700 

i 

1 

12.500 

" 

1 

18.330 

6 

J 

10.000 

1 

1 

16.660 

" 

H 

22.900 

4- 

1 

20.000 

1 

li 

20.830 

u 

« 

27.500 

M 

li 

30.000 

4 

li 

25.000 

II 

If 

32.080 

44 

2 

40.000 

' 

li 

29.160 

M 

2 

36.660 

44 

2J 

50.000 

4 

2 

33.330 

11 

2J 

41.250 

44 

3 

60.000 

' 

21 

37.500 

u 

2i 

45.830 

« 

3J 

70.000 

1 

2i 

41.660 

" 

2} 

50.310 

44 

4 

80.000 

4 

2| 

45.830 

11 

3 

55.000 

44 

4 

90.000 

1 

3 

50.000 

« 

3i 

59.570 

44 

5 

100.000 

< 

3} 

54.160 

11 

3J 

64.160 

tc 

5J 

110.000 

1 

31 

58.330 

11 

3| 

68.740 

« 

6 

120.000 

i 

3f 

62.500 

(( 

4 

73.330 

6J 

i 

10.830 

' 

4 

66.660 

11 

4J 

77.910 

21.660 

t 

4J 

70.830 

11 

4J 

82.500 

44 

li 

32.500 

1 

4^ 

75.000 

" 

4| 

87.080 

44 

2 

43.330 

4 

4 

79.160 

" 

5 

91.560 

44 

2i 

54.160 

4 

5 

83.330 

u 

5| 

96.240 

44 

3 

65.000 

H 

i 

4.376 

M 

5J 

100.600 

ii 

3i 

75.830 

232 


WEIGHT  OF  A  LINEAL  FOOT,  ETC. 


Breadth  in 
inches. 

Q 
?  A 

C  03 
C—  • 

Jifl 

Weight  in  Ibs. 

Breadth  in 
inches. 

Thickness  in 
inches. 

Weight  in  Ibs. 

c 

•£  ® 

j| 

c 

Jl 

c-  .2 

H 

Weight  in  Ib.s. 

6;> 

4 

86.66 

8 

4 

106.60 

9 

8} 

255.00 

4J 

97.50 

" 

4.1 

120.00 

" 

9 

270.00 

it 

5 

108.30 

" 

5" 

133.30 

9} 

} 

15.83 

M 

5» 

119.10 

" 

5} 

146.60 

1 

31.66 

•' 

6 

130.00 

(i 

6 

160.00 

" 

1} 

47.50 

" 

6} 

140.80 

" 

8} 

173.30 

M 

2 

63.33 

7 

i 

11.66 

u 

7 

186.60 

" 

2} 

79.16 

" 

1 

23.33 

•« 

7} 

200.00 

" 

S 

95.00 

" 

u 

35.00 

" 

8 

213.30 

" 

3^ 

110.80 

" 

2 

46.66 

8| 

} 

14.16 

" 

4 

126.60 

«' 

2J 

58.33 

28.33 

" 

4} 

142.50 

11 

3 

70.00 

u 

1J 

42.48 

M 

5" 

158.30 

11 

3J 

81  66 

11 

2 

56.66 

" 

5-^r 

174.10 

" 

4 

93.33 

u 

2J 

70.83 

H 

6 

190.00 

«• 

4} 

105.00 

11 

3 

85.00 

" 

6J 

205.80 

" 

5 

116.60 

" 

3} 

99.16 

" 

7 

221.60 

u 

5J 

128.30 

" 

4 

113.30 

" 

7i 

237.60 

«' 

6 

140.00 

" 

4J 

127.50 

" 

8 

253  .  30 

" 

6J 

151.60 

" 

5 

141.60 

" 

8J 

269.10 

" 

7 

163.30 

u 

5} 

155.80 

" 

9 

285.00 

n 

i 

12.50 

(i 

6 

170.00 

11 

9J 

300.80 

1 

25.00 

u 

6| 

184.10 

10 

1 

16.66 

" 

U 

37.50 

u 

7 

198.30 

" 

1 

33.33 

M 

2 

50.00 

" 

7} 

212.50 

" 

1} 

50.00 

ii 

2£ 

62.50 

11 

8 

226.60 

11 

2 

66.66 

tl 

3 

75.00 

M 

8} 

240.70 

" 

2^- 

83.33 

«' 

3} 

87.50 

9 

i 

15.00 

" 

3" 

100.00 

11 

4 

100.00 

«< 

i 

30.00 

" 

3^ 

116.60 

" 

4i- 

112.50 

11 

u 

45.00 

11 

4 

133.30 

" 

5 

125.00 

11 

2 

60.00 

11 

4J 

150.00 

M 

5} 

137.50 

" 

2J 

75.00 

M 

5 

166.60 

11 

6 

150.00 

11 

3 

90.00 

" 

5} 

183.30 

ii 

6J 

162.50 

" 

3i 

105.00 

(f 

6 

200.00 

" 

7 

175.00 

u 

4 

120.00 

' 

6} 

216.60 

M 

7^ 

187.50 

11 

4£ 

135.00 

1 

7 

233.30 

8 

| 

13.33 

11 

5 

150.00 

' 

7* 

250.00 

11 

1 

26.66 

11 

5} 

165.00 

' 

8 

266.60 

11 

H 

40.00 

" 

6 

180.00 

1 

8£ 

283.30 

11 

2 

53.33 

11 

61 

195.00 

1 

9 

300.00 

" 

2J 

66.66 

" 

y 

210.00 

( 

9J 

316.60 

M 

3 

80.00 

" 

7* 

225.00 

1 

10 

333.30 

" 

31 

93.33 

" 

8 

240.00 

10} 

i 

17.50 

WEIGET  OF  A  LINEAL  FOOT,  ETC. 


233 


Breadth  in 
inches. 

Thickness  in 
inches. 

Weight  in  Ibs. 

Breadth  in 
inches. 

Thickness  in 
inches. 

Weight  in  Ibs. 

Breadth  in 
inches. 

Thickness  in 
inches. 

Weight  in  Ibs. 

10} 

1 

35.00 

11 

n 

55.00 

11} 

H 

57.50 

1 

1} 

52.50 

" 

22 

73.33 

2 

76.66 

4 

$ 

70.00 

" 

2} 

91.56 

44 

2J 

95.83 

1 

2} 

87.50 

11 

3 

110.00 

(< 

3 

115.00 

! 

3 

105.00 

11 

3J 

128.30 

44 

3} 

134.10 

f 

3} 

122.50 

" 

4 

146.60 

" 

4 

153.30 

1 

4 

140.00 

M 

4J 

165.00 

" 

4J 

172.50 

! 

4J 

157.50 

11 

5 

183.30 

«' 

5 

191.60 

( 

5 

175.00 

M 

51 

201.60 

" 

51 

210.80 

1 

5} 

192.50 

44 

6 

220.00 

11 

6 

230.00 

1 

6 

210.00 

" 

6J 

238.30 

«< 

^} 

249.10 

! 

6J 

227.50 

" 

7 

256.60 

" 

7 

268.30 

1 

7 

245.00 

M 

7} 

275.00 

«' 

71 

287.50 

1 

7} 

262.50 

M 

8 

293.30 

•  c«J  :; 

8 

306.60 

i 

8 

280.00 

it 

8J 

311.60 

" 

8i 

325.80 

! 

8} 

297.50 

11 

9 

330.00 

M 

9 

345.00 

! 

9 

315.00 

44 

9} 

348.30 

M 

91 

364.10 

1 

9J 

332.50 

" 

10 

366.60 

" 

10" 

383.30 

1 

10 

350.00 

" 

10} 

385.00 

M 

10} 

402.50 

' 

10} 

367.50 

11 

11 

403.30 

M 

11 

421.60 

11 

i 

18.33 

11* 

J 

19.16 

" 

11} 

440.70 

1 

36.66 

1 

38.33 

12 

12 

480.00 

234 


WEIGHT  OF  A  LINEAL  FOOT,  ETC. 


WEIGHT  OF  A  LINEAL  FOOT  OF  ROLLED  ROUND 
IRON  IN  POUNDS  AVOIRDUPOIS. 

(480  pounds  per  cubic  foot.) 


Diameter  in 
inches. 

1 
Weight  in  Ibs. 

Diameter  in 
inches. 

Weight  in  Ibs- 

Diameter  in 
ineher. 

Weight  in  Ibs. 

Diameter  in 
inches. 

Weight  in  Ibs. 

A 

0.010 

03 

^8 

14.77 

Bf 

82.79 

8J 

206.2 

4 

0.041 

2} 

16.36 

6| 

86.52 

9 

212.2 

A 

0.091 

2| 

18.04 

5{ 

90.34 

01 

yf 

218.0 

I 

0.163 

2J 

19.80 

6 

94.26 

9V 

223.9 

iV 

0.255 

91 

"8 

21.64 

6J 

98.18 

^8L 

230.1 

I 

0.3'68 

3 

23.56 

B| 

102.20 

9} 

236.2 

A 

0.501 

3J 

25.56 

8| 

106.40 

9| 

242.5 

i 

0.655 

31- 

27.64 

6.V 

110  60 

« 

248.9 

A 

0.828 

3| 

29.82 

61 

114.90 

^1 

255.2 

I 

1.022 

3J 

32.07 

81 

119.30 

10 

261.7 

« 

1.237 

3| 

34.39 

6} 

123  .'70 

10} 

268.4 

I 

1.473 

3J 

36.81 

7 

128.30 

iQi 

275.0 

« 

1  .  728 

3| 

39.30 

n 

132.90 

10| 

281.8 

i 

2.004 

4 

41.88 

7} 

137.60 

10.} 

288.6 

« 

2.301 

45- 

44.57 

71 

142.30 

10| 

295.6 

i 

2.618 

4} 

47.28 

7-V 

147.30 

10| 

302.5 

if 

3.310 

4| 

50.10 

71 

152.20 

10^- 

309.5 

i} 

4.094 

4f 

53.02 

7-1 

157  20 

11 

316.8 

is 

4.950 

4l 

56.03 

n 

162.40 

114 

323.9 

ij 

5.885 

4J 

59.05 

8 

167.50 

lit 

331.3 

if 

6.911 

H 

62.17 

8i 

172.80 

HI 

338.7 

ij 

8.018 

5 

65.49 

S| 

178.20 

11  J 

346.2 

i| 

9.205 

5J 

68.71 

8J 

183.60 

n! 

353.7 

2 

10.470 

5} 

72.13 

a} 

189.10 

iif 

3G1.5 

2J 

11.820 

5| 

75.65 

81 

194.80 

ill 

369.1 

2} 

13.250 

5* 

79.17 

8f 

200.40 

12 

376.9 

BOLTS,  NUTS,  AND   HEADS. 


235 


BOLTS,  NUTS,  AND  HEADS. 
(Whitworth's  Proportions.) 

Weight  in  Ibs.  of  Heads  and  Nuts. 


Diameter  of 
bolt  in  in. 

Hexagonal. 

Square. 

Hexagonal. 

Square. 

Head. 

Nut. 

Head. 

Nut. 

Two 
Heads. 

Head 
&Nut. 

Two 
Heads. 

Head 
&  Nut. 

i 

0.008 

0.005 

0.022 

0.019 

0.017 

0.013 

0.044 

0.041 

A 

0.014 

0.007 

0.027 

0.021 

0.029 

0.022 

0.055 

0.048 

I 

0.029 

0.017 

0.061 

0.049 

0.057 

0.046 

0.122 

0.110 

A 

0.059 

0.040 

0.069 

0.050 

0.119 

0.101 

0.138 

0.119 

I 

0.068 

0.041 

0.104 

0.076 

0.136 

0.109 

0.208 

0.181 

0.104 

0.065 

0.157 

0.118 

0.208 

0.169 

0.315 

0.276 

0.151 

0.097 

0.246 

0.193 

0.302 

0.248 

0.493 

0.440 

0.254 

0.161 

0.362 

0.269 

0.508 

0.415 

0.724 

0.631 

0.367 

0.219 

0.551 

0.408 

0.734 

0.586 

1.102 

0.959 

i 

0.546 

0.326 

0.683 

0.463 

1.092 

0.872 

1.366 

1.146 

i 

0.724 

0  411 

1.109 

0.797 

1.448 

1.135 

2.217 

1.906 

i! 

1.060 

0.630 

1.400 

0.971 

2.120 

1.690 

2.800 

2.371 

it 

1.330 

0.759 

1.949 

1.379 

2.660 

2.088 

3.898 

3.328 

i 

1.840 

1.098 

2.625 

1.883 

3.680 

2.938 

5.250 

4.508 

if 

2.460 

1.517 

3.135 

2.192 

4.920 

3.977 

6.270 

5.327 

if 

2.920 

1.742 

3.704 

2.532 

5.840 

4.662 

7.409 

6.236 

u 

3.440 

1.991 

4.725 

3.276 

6.880 

5.431 

9.450 

8.001 

2 

4.370 

2.611 

6.384 

4.625 

8.740 

6.981 

12.77 

11.00 

2J 

6.150 

3.645 

8.858 

6.353 

12.30 

9.795 

17.71 

15.21 

2} 

8.480 

5.045 

11.91 

8.476 

16.96 

13.52 

23.82 

20.39 

2J 

11.32 

6.747 

15.59 

9.019 

22.64 

18.06 

31.18 

24.61 

3 

14.72 

8.783 

21.00 

15.06 

29.44 

23.50 

42.00 

36.06 

1 

236 


WEIGHT  IN  POUNDS  OF  HOUND  IRON,  ETC. 


WEIGHT  IN  POUNDS  OF  ROUND  IRON  FOR 


Diameter 
in  inches. 

Length  in  inches. 

N 

K 

% 

K 

% 

H 

y* 

i 

2 

3 

i 

0.002 

0.003 

0.005 

0.007 

0.008 

0.010 

0.012 

0.014 

0.027 

0.041 

A 

0.003 

0.005 

0.008 

0.011 

0.013 

0.016 

0.019 

0.021 

0.043 

0.064 

1 

0.004 

0.007 

0.011 

0.015 

0.019 

0.023 

0.027 

0.031 

0.062 

0.093 

A 

0.005 

0.010 

0.016 

0.021 

0.026 

0.031 

0.036 

0.042 

0.084 

0.126 

i 

0.007  0.014 

0.021 

0.027 

0.034 

0.041 

0.048 

0.055 

0.110 

0.166 

0.009 

0.017 

0.026 

0.035 

0.043 

0.052 

0.061 

0.069 

0.139 

0.208 

0.011 

0022 

0.032 

0.043 

0.054 

0.065 

0.076 

0.087 

0.174 

0.261 

0.015 

0.031 

0.046 

0.062 

0.077 

0.093 

0.108 

0.124 

0.249 

0.373 

1 

0.021 

0.042 

0.063 

0.084 

0.105 

0.126 

0.148 

0.170 

0.338 

0.508 

0.027 

0.055 

0083 

0.110 

0.138 

0.165 

0.193 

0.221 

0.442 

0.663 

H 

0.035 

0.070 

0.105 

0.140 

0.185 

0.210 

0.245 

0.280 

0.560 

0.840 

4 

0.043 

0.087 

0.131 

0.173 

0.217 

0.262 

0.304 

0.347 

0.695 

1.043 

it 

0.053 

0.104 

0.157 

0.209 

0.261 

0.314 

0.366 

0.418 

0.836 

1.255 

if 

0.062 

0.124 

0.186 

0.249 

0.311 

0.373 

0.435 

0.497 

0.995 

1.493 

it 

0.072 

0.143 

0.215 

0.287 

0.358 

0.430 

0.502 

0.584 

1.168 

1.752 

if 

0.084 

0.168 

0.253 

0.337 

0.421 

0.506 

0.590 

0.677 

1.354 

2.032 

if 

0097 

0.194 

0.291 

0.389 

0.486 

0.583 

0.680 

0.778 

1.555 

2.333 

2 

0.111 

0.221 

0.332 

0.442 

0.553 

0.663 

0.774 

0.884 

1.770 

2.654 

21 

0.140 

0.280 

0.420 

0.560 

0.700 

0.840 

0.980 

1.120 

2.240 

3.360 

2} 

0.174 

0.347 

0.521 

0;695 

0.869 

1.042 

1.216 

1.390 

2.781 

4.172 

2| 

0.209 

0.418 

0.627 

0.836 

1.045 

1.254 

1.463 

1.673 

3.346 

5.019 

3 

0.250 

0.500 

0.750 

1.000 

1.250 

1.500 

1.750 

1.990 

3.981 

5.972 

EXAMPLE. — Required,  the  weight  of  a  bolt  1J  inches  diameter, 
4  inches  between  inside  of  head  and  nut. 

Weight  of  bolt  =  1.39 
Weight  of  square  head  =  1.40 
Weight  of  hexagonal  nut  =  1.06  taken  as  a  hexagonal  head 

Ans.  3.85  Ibs. 


WEIGHT  IN  POUNDS  OF  ROUND  IKON,  ETO. 


237 


BOLTS,  ETC.,  BETWEEN  HEAD  AND  NUT. 


Diameter  1 
in  inches. 

Length  in  inches. 

4 

5 

6  - 

7 

8 

9 

10 

11 

12 

i 

0.055 

0.069 

0.082 

0.096 

0.110 

0.124 

0.137 

0.151 

0.165 

A 

0.086 

0.107 

0.128 

0.150 

0.171 

0.192 

0.214 

0.235 

0.257 

1 

0.124 

0.155 

0.186 

0.217 

0.248 

0.279 

0.311 

0.342 

0.373 

A 

0.167 

0.209 

0.251 

0.293 

0.335 

0.377 

0.419 

0.461 

o.5oav 

1 

0.221 

0.276 

0.331 

0.386 

0.442 

0.497 

0.552 

0.607 

0.663: 

0277 

0.347 

0.416 

0.486 

0.555 

0.624 

0.694 

0.763 

0.833' 

0.347 

0434 

0.521 

0.608 

0.695 

0.782 

0.869 

0.956 

1.043 

0.497 

0.622 

0.746 

0.871 

0.995 

1.119 

1.244 

1.368 

1.493 

0.677 

0.846 

1.016 

1.185 

1.354 

1.524 

1.693 

1.862 

2.032 

i 

0.884 

1.105 

1.326 

1.548 

1.769 

1.990 

2.211 

2.432 

2.654 

U 

1.120 

1.400 

1.680 

1.960 

2.240 

2.520 

2.800 

3.080 

3.360 

li 

1.390 

1.738 

2.085 

2.433 

2.781 

3.128 

3.476 

3.823 

4.172 

If 

1.673 

2.091 

2.510 

2.928 

3.346 

3.765 

4.182 

4.601 

5.019 

li 

1.990 

2.488 

2.985 

3.483 

3.981 

4.478 

4.976 

4.973 

5.972 

2.336 

2.920 

3.504 

4.088 

4.673 

5.257 

5.841 

6.425 

7.010 

If 

2.709 

3.386 

4.064 

4.741 

5.418 

6.096 

6.773 

7.450 

8.128 

ll 

3.111 

3.888 

4.666 

5.334 

6.221 

6.999 

7.777 

8.547 

9.333 

2 

3.538 

4.423 

5.307 

6.192 

7.077 

7.961 

8.846 

9.730 

10.610 

2} 

4.480 

5.600 

6.720 

7.840 

8.960 

10.080 

11.200 

12.320 

13.440 

2J 

5.562 

6.953 

8.343 

9.734 

11.120 

12.510 

13.910 

15.290 

16.690 

2f 

6.692 

8.365 

10.040 

11.710 

13.380 

15.060 

16.730 

18.400 

20.070 

3 

7.962 

9.953 

11.940 

13.930 

15.920 

17.910 

19.910 

21.890 

23.890 

WEIGHT  OF  MATERIALS  USED  IN  BUILDING. 


WEIGHT  OF  MATERIALS  USED  IN  BUILDING. 

(Per  square  foot  from  one  inch  thickness  to  a  cubic  foot.) 

Stones,  Earths,  &c. 


9 

tt. 

Brick. 

• 

£ 

0) 

M 

pj 

<o 

fl 

i.3 

0  9 

cS 
g" 

averaj 

SH 
0 

fi  Jj 

of  Parii 

1 

a 

1 

| 

1 

B.C 

0  C 

73 

* 

i 

9 

2 

el 
8s 

1 

o 
S 

o 

£ 

cT 
1 

1 

d 

jj> 
• 

e" 

<J 

S 

a 

5 

a 

1 

1 

1 

i 

6.58 

14.58 

8.50 

11.41 

9.33 

6.12 

9.08 

16.5 

14.08 

8.16 

8.5 

10.83 

2 

13.16 

29.1C 

17.00 

22.83 

18.66 

12.25 

18.16 

33.0 

28.16 

16.33 

17.0 

21.66 

3 

19.74 

43.74 

25.50 

34.24 

28.00 

18.36 

27.24 

49.5 

42.25 

24.50 

25.5 

32.49 

4 

26.32 

58.32 

34.00 

45.66 

37.33 

24.50 

36.33 

66.0 

56.32 

32.66 

34.0 

43.33 

6 

32.90 

72.90 

42.50 

57.08 

46.66 

50.61 

45.41 

82.5 

70.40 

40.83 

42.5 

54.16 

6 

39.48 

87.48 

51.00 

68.50 

56.00 

J6.74 

54.50 

99.Q 

84.48 

49.00 

51.0 

65.00 

7 

46.06 

102.00 

59.50 

80.00 

65.33 

42.86 

63.60 

115.5 

98.56 

57.16 

59.5 

75.83 

8 

52.64 

116.64 

68.00 

91.32 

74.66 

49.00 

72.66 

132.0 

112.64 

65.32 

68.0 

86.66 

9 

59.22 

131.22 

76.50 

102.75 

84.00 

55.10 

81.75 

148.5 

126.72 

72.50 

76.5 

97.50 

10 

65.80 

145.80 

85.00 

114.16 

93.33 

61.23 

90.83 

165.0 

140.80 

81.66 

85.0 

108.33 

11 

72.38 

160.38 

93.50 

125.60 

102.6(5 

67.35 

99.13 

181.5 

154.90 

89.82 

93.5 

119.16 

12 

79.00 

175.00 

102.00 

137.00 

112.00 

73.50 

109.00 

198.0 

169.00 

98.00 

102.0 

130.00 

Stones,  Earths,  &c. 


-^ 

i 

Granite. 

fl 

a 

o 

7j 

S 

> 

a 

0 

g 

S: 

o 

C3 

<n    . 

11 

a> 
o 

tJ 
a 

bO 

£ 

5 
1 

I? 

1 

a 

0 

o 

d 

1 

1 

73  si 

OS 

73 

& 

0? 

I 

5* 

£ 

S 

3 

1 

s 

| 

H 

1 

53 

5 

O 

0 

1 

1 

1 

s 

02 

3 

1 

6.75 

11.16 

10.0 

12.91 

10.41 

11.41 

13.75 

8.66 

12.25 

13.75 

14.08 

5.21 

2 

13.50 

22.33 

20.0 

25.82 

20.83 

22.83 

2750 

17.33 

24.50 

27.50 

28.16 

10.42 

3 

20.25 

33.50 

30.0 

38.73 

31.25 

34.25 

41.25 

26.00 

36.75 

41.25 

42.24 

15.62 

4 

27.00 

44.66 

40.0 

51.64 

41.66 

45.66 

55.00 

34.66 

49.00 

55.00 

56.32 

20.83 

5 

33.75 

55.83 

50.0 

6455 

52.08 

5708 

6875 

4333 

61.25 

68.75 

70.40 

26.04 

6 

40.50 

67.00 

60.0 

77.46 

64.50 

68.50 

82.50 

52.00 

73.50 

82.50 

84.48 

31.24 

7 

47.25 

78.16 

7Q.O 

90.37 

73.00 

80.00 

96.25 

60.66 

85.75 

96.25 

98.56 

36.45 

8 

54.00 

89.33 

800 

103.28 

83.32 

91.32 

110.00 

69.22 

98.00 

110.00 

112.64 

41.66 

9 

60.75 

100.50 

90.0 

116.19 

93.75 

102.75 

123.75 

80.00 

110.25 

123.75 

126.72 

4687 

10 

67.50 

111.66 

100.0 

129.10 

104.16 

114.16 

137.50 

86.66 

12250 

13750 

140.80 

52.08 

11 

74.25 

122.83 

110.0 

142.01 

114.57 

125  57 

150.25 

95.32 

134.75 

150.25 

154.88 

57.28 

12 

81.00 

134.00 

120.0 

155.00 

125.00 

137.00 

165.00 

104.00 

147.00 

165.00 

169.00 

62.50 

DIVISIONS    OF    A    FOOT,  ETC. 


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TABLE  FOR  COMPARING  MEASURES  AND  WEIGHTS. 


TABLE  FOR  COMPARING  MEASURES  AND  WEIGHTS 
OF  DIFFERENT  COUNTRIES. 


Weights. 


UNITED 
STATES  AND 
ENGLAND. 

PRUSSIA. 

AUSTRIA. 

BADEN  AND 
SWITZERLAND. 

FRANCE. 

Pound. 

Pound,  Z.  V. 

Pound. 

Pound. 

Kilogra'e. 

1 

0.9072 

0.8100 

0.4536 

1  .  1023 

1 

0.8928 

Same  as 

0.5000 

1.2346 

1  .  1200 

1 

Prussia. 

0.5600 

1.2346 

1.1200 

0.9999 

0.5600 

2.2046 

2.0000 

1.7857 

1 

Measures  of  Length. 


Foot. 

Foot. 

Foot, 

Foot, 

Meter.  » 

=  12  inches. 

=  12  inches. 

=  12  inches. 

==10  inches. 

=  100  Centi. 

1 

0.9711 

0.9642 

1..0160 

0.3048 

1.0297 

1 

0.9929 

1.0462 

0.3138 

1.0371 

1.0072 

1 

1.0537 

0.3161 

0.9843 

0.9559 

0.9490 

1 

0.3000 

3.2809 

3.1862 

3.1635 

3.3333 

1 

Measures  of  Surface— Square  Measure. 


Square  foot. 

Square  foot. 

Square  foot. 

Square  foot. 

Sq.  Meter. 

1 

0.9431 

0.9297 

1.0322 

0.0929 

1.0603 

1 

0.9858 

1.0945 

0.0985 

1.0756 

1.0144 

1 

1.1103 

0.0999 

0.9688 

0.9137 

0.9007 

1 

0.0900 

10.7643 

10.1519 

10.0074 

11.1111 

1 

TABLE  FOB  COMPARING  .MEASURES  AND  WEIGHTS. 

Cubic  Measure. 


UNITED 
STATES  AND 
ENGLAND. 

PEUSSIA. 

AUSTRIA. 

BADEN  AND 
SWITZERLAND. 

FRANCE. 

Cubic  foot. 

Cubic  foot. 

Cubic  foot. 

Cubic  foot. 

Cubic  meter 

1 

0.9159 

0.8964 

1.0487 

0.0283 

1.0918 

1 

0.9787 

1  .  1450 

0.0309 

1.1156 

1.0217 

1 

1.1699 

0.0316 

0.9535 

0.8733 

0.8548 

1 

0.0270 

35.3166 

32.3459 

31.6578 

37.0370 

1 

Weight  per  Unit  of  Length. 


Lbs.  per 
lineal  foot. 

Lbs.  per 
lineal  foot. 

Lbs.  per 
lineal  foot. 

Lbs.  per 
lineal  foot. 

Kil.  per 
lineal  meter 

1 

0.9342 

0.8400 

0.8929 

1.4882 

1.0705 

1 

0.8993 

0.9559 

1.5931 

1.1904 

1.1120 

1 

1.0629 

1.7716 

1.1199 

1.0462 

1.9408 

1 

1.6667 

0.6720 

0.6277 

0.5645 

0.6000 

1 

Weight  per  Unit  of  Surface. 


Lbs.  per 
square  inch. 

Lba.  per 
square  inch. 

Lbs.  per 
square  inch. 

Lbs.  per 
square  inch. 

Kil.  per 
square  cent. 

1 

0.9619 

0.8712 

1  .  2656 

0.0703 

1.0396 

1 

0.9057 

1.3157 

0.0731 

1.1478 

1.1041 

1 

1.4526 

0.0807 

0.7902 

0.7601 

0.6884 

1 

0.0556 

14.2223 

13.6811 

12.3910 

18.0000 

1 

16 


242 


RESISTANCE  TO  CROSS-BKEAKING. 


RESISTANCE  TO  CROSS-BREAKING. 

To  Cut  the  Strongest  and  Stiffest  Rectangular  Beam  from  a  Log, 
Fig.  308.     (Strongest.) 


The  diameter  aa  =  d,  divided  into  three  equal  parts,  with  per* 
pendiculars  J  d  from  a  erected  thereon,  intersecting  the  circle  at 
b,  will  give  section  for  greatest  capacity. 


Fig.  309.    (Stiffeet.) 


The  diameter  aa  =  d,  divided  into  four  equal  parts,  with  per- 

rndiculars  J  d  from  a  erected  thereon,  intersecting  the  circle  at 
,  will  give  section  with  least  deflection,  but  less  capacity  than 
Fig.  308. 


INDEX. 


Area,  circumference,  and  cubic  contents  of  circles 218 

Axis,  neutral 4 

Bars,  tie  rods,  &c 181 

resistance  of,  to  tearing 2 

Beams,  capacity  and  strength  of 29 

of  rolled 39 

of  cast-iron 57 

TFof  rolled  l-shaped 39 

and  strength  of  parabolic  arched 153 

cast-iron 53 

iron  ties,  struts,  and 3 

sloping  rafters  and 102 

strains  in  trussed 122 

horizontal  andsloping 188 

strength  of  wooden 88 

Bolts  and  nuts,  dimensions  of. 187 

nuts,  and  heads 235 

Boom  derricks,  strains  in 114 

Booms,  strains  in  trusses  with  parallel 126 

Bow-string  girders 147 

Bridges,  static  and  moving  loads,  of  wrought  iron 192 

Camber 2 

Capacity 2 

and  strength  of  beams 29 

W  of  rolled  l-shaped  beams 39 

of  rolled  beams 41 

of  cast-iron  beams 57 

and  strength  of  parabolic  arched  beams..... 153 

Cast-iron  beams 3,  53 

Center  of  gravity  of  planes 202 

Circumference,  area,  and  cubic  contents  of  circles 218 

Columns,  pillars,  and  struts,  strength  of 110 

Composition  and  resolution  of  forces Ill 

Compound  strains  in  horizontal  and  sloping  beams 188 

Compression 1 

Compressive  strain  and  pressure  on  supports 102 

Contraction  and  expansion 4 

(243) 


244  INDEX. 

MOB, 

Constants  for  strain  in  trusses 117 

roof  trusses 174 

Connections  in  iron  construction,  joints  or 184 

Cross-breaking 2 

and  shearing,  resistance  to 29 

Crushing,  resistance  to 103 

direct 1 

Deflection 2 

Derricks,  strains  in  boom 114 

Dimensions  of  bolts 187 

Divisions  of  a  foot,  expressed  in  equivalent  decimals 239 

Expansion  and  contraction 4 

External  forces .T^J/i 

Factors  of  safety 29 

Forces  external .,  ..r  1 % 

internal 1 

composition  and  resolution  of. Ill 

parallelogram  of Ill 

Frame,  strains  in  polygonal 154 

Functions,  trigonometrical 207 

Geometry 197 

Girders,  strains  in  parabolic  and  bow-string 147 

Gravities  of  materials,  specific 224 

Heads,  nuts,  and  bolts 235 

Horizontal  and  sloping  beams,  compound  strains  in 188 

Howe  truss 129 

Inertia  and  resistance  o  various  sections,  moments  of 5 

Internal  forces 1 

Iron  beams,  capacity  of  cast 57 

cast 53 

bridges,  static  and  moving  loads,  of  wrought 192 

construction,  joints  or  connections  in .-. 184 

ties,  struts,  or  beams 3 

Joints  or  connections  in  iron  construction 184 

Lattice  truss 139 

with  vertical  members 131 

Longimetry  and  planimetry 197 

Materials,  &c.,  strength  of 26 

Miscellaneous , 195 


IffDEX.  245 


Modulus  of  rupture 4 

Moment  of  inertia  and  resistance  of  various  sections 5 

Moving  loads,  weight  of. 191 

Natural  sine,  cosine,  &c 306 

Neutral  axis 4 

Nuts,  heads,  and  bolts 235 

dimensions  of... 187 

Parallelogram  of  forces..... Ill 

Parallel  booms,  strains  in  trusses  with 126 

Parabolic  arched  beams,  capacity  and  strength  of. 153 

curved  trusses,  strains  in 147 

Planimetry,  longimetry,  &c 197 

Pillars,  columns,  and  struts,  strength  of 110 

Pins,  &c.,  in  tie  bars 185 

Polygonal  frame,  strains  in ,.  154 

Pressure  on  supports 100 

compressive  strain  and 102 

of  snow  on  roofs 178 

of  wind  on  roofs 180 

Rafters,  &c.,  sloping  beams _. 102 

Reactions  of  supports 100 

Resistance  to  direct  crushing 1 

of  bars,  &c.,  to  tearing 2 

to  cross-breaking  and  shearing 29 

crushing 103 

Resolution  of  forces,  composition,  &c Ill 

Rolled  beams,  capacity  of. 41 

l-shaped  beams,  capacity  of. 39 

Rods  and  bars,  tie 181 

Roof  trusses..., 3 

strains  in 156 

constants  for  strains  in 174 

Roofs,  pressure  of  wind  on 178 

of  snow  on 180 

Rupture,  modulus  of. 4 

Shearing 2 

and  cross-breaking,  resistance  to 29 

Sloping  beams,  rafters,  &c 102 

and  horizontal  beams,  compound  strains  in 188 

Specific  gravities  of  materials 224 

Static  and  moving  loads  of  wrought-iron  bridges 192 

Strength  of  materials 26 

wooden  beams 98 

columns,  pillars,  and  struts 110 


246  INDEX. 

PAGE. 

Strength  of  beams,  capacity,  &c 29 

Strains  in  frames 112 

boom  derricks 114 

trusses 115 

trussed  beams 122 

trusses  with  parallel  booms 126 

parabolic  curved  trusses,  or  bow-string  girders....  147 

polygonal  frame 154 

roof  trusses 156 

constants  for 174 

trusses,  constants  for 117 

Strongest  and  stiffest  rectangular  beam  from  a  log,  to  cut  the..  242 

Struts  and  beams,  iron  ties 3 

Supports,  reaction  of. ...  100 

compressive  strain  and  pressure  on 1 02 

Table  for  comparing  measures  and  weights 240 

Tearing,  resistance  of  bars,  &c.,  to 2 

Tension 1 

Tie  rods  and  bars 181 

Trigonometrical  functions 207 

formulas 205 

Truss,  Howe 129 

Warren 132 

Whipple 144 

lattice 139 

with  vertical  members 131 

Trusses  parallel  booms,  strains  in 126 

parabolic  curved,  or  bow-string 147 

constants  for  strains  in  roof 1 74 

constants  for  strains  in 117 

strains  in 115 

roof 156 

Trussed  beams,  strains  in 122 

Warren  truss , 132 

Weight  of  moving  loads 191 

static  and  moving  loads  of  wrought-iron  bridges...  192 

a  lineal  foot  of  flat  or  square  bar  iron 229 

rolled  round  iron 234 

materials  used  in  building 238 

superficial  inch  of  wrought  and  cast  iron 227 

rolled  round  iron  for  bolts 236 

heads  and  nuts 235 

per  square  foot  of  metals 228 

Whipple  truss 144 

Wooden  beams,  strength  of. .  98 


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